Out for lunch Saturday with Better Half's Mum and her Better Half, when BHM'sBH declares "I hate it when English people take on American corporate jargon." I expected he was talking about thinking outside the box and investing in excellence. The latter is one of my pet peeves, as well as the current call-to-arms/pens by my university's management. ( I also hate that my university has a management team now instead of an administration.)
But no, BHM'sBH instead went on to talk about how he can tell who works for an American company when they open their mouths at Starbucks and say "Can I get a..." instead of "Can/May I have a...". If you ask to get a coffee, by his reasoning, you're asking to come (a)round to the other side of the counter and fix yourself a coffee. BH agrees with BHM'sBH that in the context of ordering a coffee can I get a means 'can I get myself a'.
Checking out can I get a on Google UK, some of the examples are:
can I get a qualification?
Can I get a regular health check at my GP surgery?
can i get a gmail invite?
Can I get a refund of unused portion of a season ticket...?
Can I get a refund on my parking permit?
Can I get a business grant to start up a new business in the Harrogate district?
Most of these are from FAQs, and are sincerely about ability ('Am I able to get a refund? How do I do it?'), rather than requests for things to be given. That's not surprising, as writing Can I get a decaf latte? on my website is unlikely to result in a hot beverage showing up beside me. (That's what yelling to BH in the kitchen was invented for.) The gmail invite example is the only one that stands out as a UK-located (but not necessarily UK) person requesting something using can i get a.
BrE speakers have no problem with saying I got a coffee on my way to work, meaning 'I took a coffee away from someplace where I ordered it'.
In can I get a, get is the converse of give:
Can I get a coffee (from you)? = Can you give me a coffee?
But it's not that the get/give opposition is American-only: on UK Google, one gets twice as many hits for get presents as for receive presents. Since one typically doesn't go and get one's presents from the giver ( one tends to passively receive them), it's clear here that British folk have the lexical/logical wherewithal to understand can I get a as a request.
I think is the real problem is that one learns early the 'polite' ways to ask for things, and this way is not in the British canon of polite requests. While get can mean a passive action of receiving, it also has other senses that are closer to take--which probably colo(u)rs people's perceptions of get's connotations. So, if you're brought up on saying can I have a, then can I get a might sound a little more greedy/impolite.
But why doesn't it sound less polite to American ears? (Especially if it's a relatively new locution there too?) Three possibilities, which don't rule one another out:
1 - Perhaps it does sound less polite to Americans too. To me can I get a sounds a little more brusque and self-cent(e)red than can I have a. (But maybe I've been influenced by my surroundings.)
2 - Perhaps it is slightly less polite in both dialects, but it's less important to sound "polite" in America. (The word polite is a bit loaded here. I'm using it to mean something like 'indirect/genteel'.) The US is known for its solidarity politeness system and for its individualistic culture. In a solidarity culture, one wants to act as if everyone's on the same social level. The UK is historically a deference culture, in which people's societal roles are more distinguished and great pains are taken not to inflict oneself on others unnecessarily. The UK has been shifting toward solidarity styles since at least the second world war, but is still not as far along that path as the US. The importation of (and unease with) can I get a may be a symptom of that shift.
3 - Or perhaps it's just that anything that sounds American grates on British ears and sounds less polite, just by association with Americans and stereotypes of Americans.
I think it's probably a combination of all of these.