I have noticed the tendency of replacing “may I” with “can I” in the spoken English and in some movies. 

However, in formal spoken or writing or on serious TV channels the usage is correct.

Is there any particular reason, or some nuances which I possibly cannot get, that make people to use “can I” instead of “may I”? 

On the other hand, sometimes, the usage of “can I” instead of “may I” leaves room for jokes such as: 

“Can I get you a drink”?

“I don’t know, can you”?