I will agree with you that AI do not have feelings, but I would ask: how can you measure what has a soul and what doesn't have a soul? Some religions believe that everything in the universe has its own soul. I don't really consider a soul to be a strong factor to consider.

My question is if an AI is ever capable of having its own feelings, should it have rights?

I would say that animals do have rights to some extent. They are protected from things like dog fights and abuse, and there are rules stopping habitat destruction for endangered species.