For you to deserve to be called a ... you should

What is it called when someone tries to manipulate someone into behaving a certain way by questioning core identity characteristics valued by the other person, if the other person doesn't behave as the manipulator wants.

It can come into two forms:
1)You are not a X if you don't do Y.
2)You don't deserve to be a X if you don't do Y.
when Y is not a prerequisite for X.

"You don't deserve to be called a man if you don't want to fight".
"If you want to be called a human, you should respect others" 
"You don't deserve to be a blonde if you don't perform great fellatio"

It reminds me a bit of stereotyping (the fallacy) but it doesn't cover all the cases where this tactic is used. For example it can be used even when the claim is not a somewhat accepted stereotype.

Then it reminds me of the  definist fallacy,  which doesn't cover though the "you don't deserve to be called" subtype which in turn is a bit similar to no true scotsman but not quite exactly.

Is there a more specific name for that kind of rhetoric?

I agree with all of your mentions of fallacies, especially no true scotsman . Who says one is not an X if they don't do Y?

Could also be a form of gaslighting... making people think that it is the case that you must do x to be y when it is not the case. Other than that, it just appears to fall under rhetoric and possibly manipulation.

Well, the main problem is that the argument is not against a claim as it is directed at the opponent's self-confidence.

"If you really believed what you were saying you'd think and act like this - but you don't!" 

So, yes, No True Scotsman is on the money, but I would suggest that argumentum ad fidentia is a close second. 

Try the no true scotsman  fallacy.