In this case, 내/제/네 is not short for 나의/저의/너의. It's just that 나/저/너 becomes 내/제/네 when used with 가.
It's a rule that exists without any reason, or at least I can't work out. In 경기도 and 서울, people will say 너가 and 니가(less common). In other cities, people say 니가 in general.