I wrote two posts about this, but decided they are really probably more than you want to know.
I think it boils down to two main points. The idea of a soul isn't some random thing, it's from platonic or neoplatonic philosophy, and it relates to the distinction between form and matter which plays an important role in the coherence of that system, it's metaphysics and epistemology. A soul is just the form of a living thing, its mathematical reality which makes it more than a differentiated lump of something. Platonism is still a very common philosophical position among philosophers, mathematicians and scientists, and it underlies a lot of the major religions as well.
Obviously not everyone is a platonist and you can argue against it, but to speak as if it's some kind of fly-by-night, obviously silly way of thinking, or that people just believe it because of some random supernatural reason unconnected to their more sensible ideas, is a bit dismissive.
Gender identity, on the other hand, is an idea that seems to be a bunch of mutually exclusive claims without much more than some poorly thought out languaguge based arguments attached, which even the people that push it can't explain in most cases, and with associated scientific claims that aren't evidentially supported.
There is an interesting difference though between the use of the word soul in these two instances. If we accept the soul type language for gender identity - and tbh I am not sure we should, I think that is largely loose talk - they can't mean the same thing. In the Christian view, a soul could not be in the wrongly sexed body as such because it is the soul that gives the body the shape that it has - it is that shape, really. The two are a unity. The idea that you could have a male soul in a body that was female is just incoherent from that perspective.