Why is "can I get" replacing "could I have"?

I noticed the other day when serving the public that when asking for something, people were saying "Can I get an xyz, please". The previous time I had such a job it was "Could I have an xyz", or "May I have..." if they were being very polite.

Does anyone know when this trend started and what caused it?

I know the question is tagged “British English”, but for some historical perspective in American English, this kind of request, formulated as “Can I get”, is not exactly some new invention. Here is a passage from a novel written in 1859:

“Have you any work, sir?” was repeated in a still lower and more timid voice than that in which her request had at first been made.

“Yes, we have,” was the gruff reply.

“Can I get some?”

And here’s one from a 1906 novel:

Then our hero entered the place. “Can I get a room here for the night?” he asked of the clerk behind the desk.

And here, from a 1938 Time magazine article:

At one ticket counter a soft-voiced woman had the ear of a clerk. “Can I get two tickets to Boston?” she inquired. The clerk asked who the other passenger was.

“He’s in a hearse outside,” the woman explained.

As a British observer I can confirm this is an Americanism and is being taken up by young, non-discerning British people who are being influenced by American T.V.

In Britain 'get' has a similar meaning to 'fetch' so a customer in a shop asking 'can I get...' would be suggesting that they actually fetch the article from wherever it is which is surely the duty of the shopkeeper: the shopkeeper 'gets' the article so that the customer can 'have' it.

'Could I have...' is surely the correct way for a customer to request an article or service (at least in Britain).

Is the question about the difference between 'Can I ...' & 'Could I ...' and 'May I ...' & *Might I ...'?

I will not address the difference between can and may as this has been discussed in the comments on the question (they are now part of idiomatic English for better or worse).

May and might are known as modal auxiliary verbs. According to the Oxford Guide to English Grammar, 'with modal verbs, we express ideas such as actions being possible or necessary' (p. 78) and 'the modal verbs are will, would, shall, should, can, could, may, might, must, need, ought to, and dare' (p. 113). In most cases, modal auxiliary verbs do not have tenses, and The Cambridge Guide to English Usage asserts that 'May I have a cup of tea?' and 'Might I have a cup of tea?' are grammatically the same, with the second '[seeming] overanxious to let the other party determine your right to the simplest of drinks' (p. 340).

In short: 'Can I ...' & 'Could I ...' are part of idiomatic English, but I wouldn't recommend them. The difference between 'May I ...' & 'Might I ...' is not grammatical but stylistic. All things considered, it is probably better to use 'May I ...' rather than any of the other alternatives as it is definitely correct and does not show overanxiousness.