Sounds like typical boy competition to me. If the cousin was showing off, then he should expect to be challenged. This sort of thing happens in adulthood as well--for example, I was in my gay group last year, mouthing off about something (I can't remember what), and a couple of the guys took me down a peg. I had second thoughts and retracted, and we went on with our conversation. No big deal. They were quite right to say something; I was being obnoxious.
I should also point out that if the roles were reversed, it wouldn't be a "fair fight," so to speak. From what I have read, nobody was trying to say that the cousin isn't a real man, but trans men are told that pretty consistently. Apples and oranges. Anyway, if the OP were going overboard and other men were around, he would probably be challenged as well.