A - So, how come he gets to give you a gift and I don't?
B - Because I have nothing to give back to you.
My question is: why to use the verb "to get" in this case? I think "can" is more suitable. I cannot catch the meaning of this "get"...
So, how come he gets to give you a gift and I don't?
"to get" has this meaning too. Here it means "finds a way to", "can", "is able to", "gets a chance to", "you accept gifts from him but not from me". I guess you got the picture. It's about jealousy.
We use "get to" to express something that is a positive experience or a good thing. In this sentence "he gets to give you a gift" the inference is that it is something that he wants to do and is allowed to do it and there is some positive reward in doing it.
Let me give you a different context comparing "have to" and "get to".
I love my job. I get to travel a lot (the inference here is that I love traveling)
I hate my job. I have to travel a lot (the inference here is that I hate traveling).
Hope this helps but I don't understand the full context.
Today I have come across this sentence: I'd like to get to you know you a little better.
Does "get" mean "to manage", "to be allowed" too, as in the previous sentence?
"Be allowed to." You might want to get to know a woman better, but she may not want that.
Here "to get" is like "to get a chance", "to get an opportunity", "to make it possible".