I think that as a Catholic, if he accepts Catholic teaching on the matter (if I have understood it correctly) he wouldn’t believe that a soul could be in the wrong body. Catholic teaching as I understand it is that the soul and body form together, intimately connected and not that the body is a vessel for the soul but that neither exists without the other.
So I don’t think that the Catholic view would be that gender is a sort of soul. I could be wrong, a practising Catholic or even better a Catholic theologian may be along to explain it better if we are lucky.
I heard an Orthodox priest say that every living thing has a soul and describe it as the thing which makes something alive. I think this may differ from the Catholic view.
People who believe in gender identity have quite a different view of the soul than either the Roman Catholic or Orthodox Christian perspective, I would say that the gender identity view of the soul is closer to Gnosticism which is considered a heresy by both Catholics and Orthodox Christians. Gnosticism does believe that the body and soul are separate and the soul is the true person and that the goal of the soul is to escape the body.