Can I have...?
Languages have many ways of addressing other people politely. Many languages like German or Spanish avoid addressing a person too directly by using the third person, as if they were talking to someone else ("¿Le gusta...?", "Möguen Sie...?"). In Japanese it can be considered very rude to call someone "you" (anata).
Obviously, in English you don't use the third person (imagine asking someone "Can he help me?" instead of "Can you help me?") and using "you" is perfectly acceptable, but to show deference, it is common practice to avoid asking other people to do things for you , which is considered too direct, and ask whether you can have something instead (alternatively, you can use the polite could, would...) Spanish does not use this approach, so asking literally "Can I have...?" sounds a bit like "Can I possess...?" It is weird. In Spanish you do ask people to do things for you (instead of asking whether we can have something from them), but we can use "usted", the conditional tense and other mechanisms to make the sentence more polite.
The following translations are taken from the Collins dictionary. Notice how "Can I have" is systematically replaced with "dar" or other verb where the other person will actually perform the action:
• can I have your name please? ¿me da su nombre, por favor?
• can I have Personnel please? ¿me puede poner con Personal, por favor?
• please can I have my book back? ¿me puedes devolver el libro, por favor?
• can I have a pencil please? ¿me puedes dar un lápiz, por favor?
These are taken from the Oxford dictionary:
• can I have two salads, please? ¿me trae dos ensaladas, por favor?
• can I have the ring back? ¿me devuelves el anillo?
This is from the Chambers Harrap:
• can I have a beer and a brandy, please? -> ¿me daría or pone una cerveza y un coñac, (por favor)?
This is from the Velazquez dictionary (surprisingly, the same example as in the Oxford!!!!)
• Can I have two salads, please? -> ¿me trae dos ensaladas, por favor
The only cases where we actually say "Can I have...?" is when you are really asking whether having something is allowed, not when you are asking people to give you things. E.g.
• Mummy, can I have a dog? - Mamá, ¿puedo tener un perro?
I can't tell whether there are regions where Spanish speakers have adopted the English formula to express politeness, but I have checked with two friends of mine, one from Mexico and another one from Argentina, and both suggest sentences like "¿Me da...?" and think that "¿Puedo tener...?" to order things in a restaurant sounds extremely weird. I have also checked a few threads in other fora (e.g. WordReference), and people from other Latin American countries also discard "¿Puedo tener...?" as a weird way to order politely. Maybe other native speakers can provide further insights, but be warned: it sounds awfully weird in many countries.
17 Answers
On a related matter, I once read a nice summary of different ways people order food in a restaurant.
Direct: Give me a slice of pizza.
Direct polite: Please give me a slice of pizza. (all forms can optionally include please)
Declaring a desire: I want a slice of pizza.
Asking permission: Can I have a slice of pizza?
Experimental: I'll try a slice of pizza.
Conditional: Would you get me a slice of pizza.
Asking approval: How about a slice of pizza?
Tentative: I might have a slice of pizza, if you have it.
Thief: I'll take a slice of pizza. (thanks, Stadt!)
Commanding: Bring me a slice of pizza.
.
I may have missed a few; I'm sure there are more.
It´s possibily Spanish influence over English, but this thing of ordering something in a restaurant by saying "Can I have ...." has always bothered me. "Can I" is asking "if I have the ability" and as long as you are paying, of course you have the ability to receive anything on the menu.
It seems that lately people in the English-speaking world have fallen victim to this ridiculous notion called "political correctness," which many times takes things to extreme.
And with asking permission, I prefer to say "May I ......?" Were you to ask, "Can I ....?" someone could respond by saying "Yes, you can" by meaning "Yes, you have the ability" while not giving you the permission you are seeking.
It´s best to be direct and precise and leave the "political correctness" to the nicey-nicey, trendy folks. This can be done while being polite at the same time.
Even within the English language, rules or acceptable ways of asking for things vary from country to country and region to region.
I was once in a South Carolina restaurant (forgive me southern friends) where the man in front of me, after what seemed an eternity of contemplation, said:
"Hi, how y'all doin? You know, I'm fixin to have me a slice of that there square pizza. (It was a rectangular Sicilian pizza, but we'll forget about geometry for now.) I can't seem to recall what the "I"talians call it? Anyway, if y'all could give me a slice of that there pizza, that would be great! Y'all have nice day now, you here.
Where I live, anyone standing behind this person would be ready to choke him. Here it is perfectly polite and actually preferred to say something short and simple like:
"Yo, give me a slice to go. Have a good one."
Julian said:
but this thing of ordering something in a restaurant by saying "Can I have ...." has always bothered me. "Can I" is asking "if I have the ability" and as long as you are paying, of course you have the ability to receive anything on the menu.
It seems that lately people in the English-speaking world have fallen victim to this ridiculous notion called "political correctness," which many times takes things to extreme.
Your answer addresses something that has been driving me crazy with my 16 year old daughter and her friends. Over the Christmas holidays, we went away for a few days my daughter brought her friend. After ordering two to three meals a day for five days, I had had enough. Each time the waitress took their order, they sheepishly said "Can I have....blah, blah, blah??? I am emphasizing the question marks because I can't adequately replicate the tone of their voices here. They actually sounded as if it weren't too much of a bother, or if the chef in the back was in a good enough mood....perhaps, maybe, if it is o.k., they could per chance have a dish of pasta, etc.
I finally had to say something. Although I commended them for their attempt at being polite, I tried to instruct them on asking in a friendly but more assertive way. After all, we are the paying customers. I think I was going more after a life lesson. I just wanted them to know that in general, you don't have to apologize for your preferences or choices in life. I applaud you for this reference to "political correctness" Julian, because in the U.S., it is becoming a bit ridiculous. Also, when they are dating boys, I want them to be able to express what's on their mind without feeling the need to ask for permission or to sound apologetic in their approach.
By the way, until they are comfortable with the assertiveness training....can they take your dog with them on their first couple of dates? 
When I was a child we had to ask before we left the table after a meal. If we said ''Can I leave the table?" My father would say "Yes, you can but you may not, stay there" We soon learned to use the more correct ''May I? ".
I do agree may is rarely used now though.
l always thought that ''¿puedo tener ..?'' is used a lot as a straight translations of a polite way to ask for things
Well, not really, lovely.
¿Puedo tener una cerveza?
This sounds like you are asking your father if you can have (not drink!) a bottle of beer in your room, like "Puedo tener un perro?"
That´s good because I have been wondering how say this for awhile. I instinctively that puedo tener was not correct. Useful for ordering things.
Lazarus, can you read my mind through the world of the internet lol?
After my question yesterday about "Will you please" I have been trying to think through ways of politely making requests all day today!
Another great insight into the Spanish way of thinking!
Muchas gracias mi amigo 
My mind is slipping
It was tendré. Still though, is tendré not acceptable either? - bafalck
Not in Spain, we would say: tomaré, me tomaré....
l always thought that ''¿puedo tener ..?'' is used a lot as a straight translations of a polite way to ask for things ''Can l have?'' but never thought that it can be replaced with the third person command, so this is a new information for me, thank you.
You provided two examples of ways asking for things politely in your post, so l'm adding another one, in Arabic, there's no third person command, but we rather do like it's done in English, we simply use the ''You - tu'' form (there is no ''usted'' in Arabic, just like English), but we add words like ''Please'', plus some other words which show the respect to the other person, but many of those words are translated as ''you'' in English.
I watched it awhile ago so I could be wrong, but wasn't there a lesson where Paralee suggested tendría when ordering in a restaurant? I haven't ever asked like that yet so I don't know if it is acceptable, but from what you are telling us, we shouldn't?
I struggled with not being able to use 'puedo tener' when first learning Spanish but soon got accustomed to phrasing my want in more natural ways.
I have never heard, not that that means that it has not been uttered within earshot, 'puedo tener' being used acceptably in the context to which Lazarus pertains throughout all my travels in Latin America.
This is a very useful post Lazarus, thank you. I remember once going into a papelería in Mexico to pick up some photocopies with my friend Normita, and I said to the clerk, "Puedo tener las copias por favor", y Normita very sternly told me that, although that was understood, I should never again say it like that! She suggested just what you suggest: ¿Puede darme las copia?, or even, Pudiera darme las copias? I have forgotten many things in retaining the modest Spanish I possess, but that is not among them.
Re: the English way of requesting
I am virtually incapable of saying "Can I have a ............?"
Maybe it's because as children we had it drummed into us that "May I" was the only correct way to ask if one was permitted to have/do something, or maybe it's something to do with what Nicole says - that when in a restaurant I don't feel I'm there to ask permission but to request food!
For me, it feels right to say:
"I'd like a ........please." It's perfectly polite and doesn't sound as if I'm apologising for being there!!
Good point to raise as always, Lazarus. 
Por cierto, ¿qué te parece la frase "Sería tan amable de traerme ......?"
A mí me suena muy bonita pero hoy en día quizá parezca muy solemne.
Gracias.
I'm not sure about this 'may I have' banter.
'May I have some Squeezy Cheezy Peaz please' could equally be met with the response 'Why, of course you may sir, youneed only request that I bring you a portion and it would be my very pleasure to bring them to you.' as could the request 'can I?'.
Neither of these expressions are actually requesting that something be done, rather enquiring as to the possibility of it being attainable.
