what is the exact meaning of 'would'?

"I wouldn't call you a tomboy" is a polite/gentle way of saying "I disagree with your assessment that you are a tomboy."

The implied sense is "If somebody asked me to describe you, I wouldn't call you a tomboy"<—"I could be wrong, and you are of course the ultimate arbiter of who and what you are, but what I know of you would lead me to describe you to somebody who asked me as something other than a tomboy."

In English, such conditional verbs are often a marker of politeness/consideration.

"And I won't" is a very subtle meta-linguistic joke, based on a purposeful misunderstanding of the implied premise of "if somebody asked me to describe you" as replaced by "if I had the opportunity to call you a tomboy right now." "If I had the opportunity to call you a tomboy right now, I wouldn't call you a tomboy. And I do have the opportunity to call you a tomboy, so I won't call you a tomboy."