When we use Korean, if the object is very clear, then we don't use it. So in this sentence, you would say that to your close friend so the object is very obvious so you can skip '내, 나의'

"생일축하해 내 웬수야" sounds okay. But usually I say like "생일축하해 웬수야" because this one is sounds more natural. :) Hope it helped.

Hello my name is Alevtina. I study English. Do you mind if I ask some questions? What is your name?