'Can I get' is appalling and sounds so rude. I'm sure it is an Americanism. No one said it in the 80s in Britain.
I had no idea it was unusual until I read about it on mumsnet. Over the past few months, I have listened intently in the queue at coffee shops, and when out with groups of people, and almost everyone says 'can I get' which is why I was wondering if it is regional. I've never really heard anyone say 'please may I have', it's just not something that is said in my part of the world, it would sound really out of place. I'm in N Ireland which is why I had been wondering if it might have originated in Scotland, but obviously I was wrong on that.
But it's definitely not a modern thing. My octagenarian mother would say 'can I get a coffee please' if she was in a cafe.