Thinking in terms of context rather than grammar, the person is stressing that they like someone who thinks otherwise. So the conversation would be:
Girl: You don't really like me
Boy: I DO, I DO like you ... really (ie., it's true despite what you think)
Here it seems that 'really' doesn't mean "a lot" as in "I really like you" = "I like you a lot"
but rather, "it's true/a fact that I like you".
So, if the guy wants to contradict the girl, and then emphasise that he likes her a lot he would say:
Boy: "I *do* like you, really. I *really* like you.
The first and the second sentence do not mean the same:
(1) It is a true (real) fact that I like you, depite what you think;
(2) In addition, I like you a lot.
To avoid confusion, it would be better to say (what would mean the same):
"In fact, I *do* like you. I like you *a lot*"
It's important where you put the *stress*.
The point is I guess that "really" is ambiguous between
(1) "a lot", eg., the movie was really good, and
(2) "it's true despite what you think/believe", eg., you don't believe me, but it's true, he really kissed her at the party.