Interestingly enough, I was listening to a podcast from “The History of English” this very morning on this topic.
Very enlightening, and the whole me/my phenomenon was discussed at length. It seems “me” as a synonym for “my” is found far more frequently in those regions settled by the danish vikings (Northumbria, etc.) than it is down south and west, like in Wessex.
OK, I just asked my kid to check it on Beatles Rock Band and they also say “my”, as well as every set of lyrics I could find online.
Is Ringo from an area of the UK where he would be expected to say “me”? I have an English friend, and he uses it very rarely and, it seems, mostly ironically.
“No, it’s just me face.”
~Ringo Starr about comments in the early days that he looked sad.
Now having said that, Can’t Buy Me Love is still not saying ‘Can’t Buy My Love’.
I think i is just leaving (for) out. Can’t buy (for) me love.
I have been told to "buy me beer " here in Texas many times !
I’m from Liverpool and we do indeed say me instead of my. although not everyone does it and not all the time
where are me shoes? I forgot me bus-pass
my wife is not from the area and finds it hilarious 
It’s a class thing as well as regional. AIUI John Lennon, however much he might have pretended otherwise, was “brought up proper” in the kind of lower middle class milieu that could get paranoid about sounding “common”.
And yes, it can be used to stereotype Northern working class people; once upon a time, there was much mockery that, in the soap Coronation Street, the mardy teenager Tracy Barlow was always grumping off upstairs “ter play mi tapes”, until one day she did it and wasn’t seen again for fifteen years or so, when she re-emerged as a femme fatale. Or, in The Fast Show, there was a repeated sketch where a very dull man tried to join in a conversation with some joke or irrelevant remark that fell flat, and in the silence he would say “I’ll fetch me coat…”
nm
“Tie me kangaroo down, Sport” If yer not old, it’s a song from the 60’s…