Anthony was brought up by Kay since he was a young boy, with possibly very little influence from his father or his father's family. After her bad experiences with the Corleones, Kay probably actively encouraged him to be as different from them as possible, encouraged him to be un-macho and take an interest in music and art etc etc. Perhaps she hoped (knowing how similar he was to the young Michael) that he would turn out how she hoped Michael would've turned out if he hadn't got involved in the family business.

Don't know if Anthony is gay or not, don't think the character was developed that far. Would have been an interesting storyline though, wonder how that woul've gone down with Michael!!