Probably gonna get banned immediately but I can at least share some insight before I'm kicked out. I'm trans male. So when I popped out the doc said "girl" and handed me to my parents. (Just preemptively clearing up any term confusion). "Feeling like a man/woman" or any variation is more or less an attempt at explaining gender dysphoria. I'll try to explain it a little more clearly.

Imagine if you looked down and one of your legs was the wrong way around. No pain, you can still function, your leg is just backwards. Would you not feel a profound sense of wrongness about your body? Your brain knows your leg absolutely should not be like that and yet it is. Every time you try to walk you are reminded of the existence of your backwards leg.

My brain reacts this way to my female secondary sex characteristics. As for why you don't "Feel like a woman", your gender ans sex match. So your brain doesn't have this reaction to anything about your body. It's not something forefront in your mind, and thus, you don't "feel" anything. Same reason you can't taste your own tongue, really.