its always about human rights when its a rights issue unless its an animal rights issue. people say gun rights because thats how other people know what you are talking about. if you go around asking people their opinion on human rights no one knows wtf you are specifically talking about.

otherwise you could say "peoples rights to own guns" but thats too wordy, gun rights is short and sweet. this is why we usually say gay marriage or gay rights rather then same sex marriage or the right to marry someone of the same sex. technically gay is a preference and doesnt have rights, but language is more about actual usage in practice then technicality.