Indeed - a great big part of "why do Americans speak funny" is that they speak old-fashioned. WE've changed our language to accommodate our neighbours and colonials.
They have slightly differing implications - "may" is a definite request for permission, whereas "can" could instead be asking about capability (particularly if there's no "please").
e.g. "May I put my whole fist in?" vs "Can I put my whole fist in?"
"Can I get" is certainly an Americanism however, as mentioned previously (on this thread?) this is another example of Old English being preserved by the colonials and the language evolving in Blighty.
Indeed, only this week I heard Joe Biden imploring his fellow Americans "if you haven’t gotten...
That is probably from Les Dennis' impression of Mavis from Coronation Street, and the correct line is "I don't really know".
Fun fact* - the actual Mavis character only said that line once in her whole 26 year run.
I have heard "I don't rightly know" used in real life, but then I am from...