This is probably one of the most coherent explanations of the word "gender" (in relation to gender identity) that I've ever read.
I still don't believe that I, or anyone else, has a "gender identity" (instead I recognise that we all respond to sex-based societal expectations and limitations in different ways) but your description of gender as an explanation for this response reads really well.
Ultimately though, the idea of gender identity all comes down to each of us answering one question: "which set of sex-based stereotypes fits you best?" If it's variable, you're gender-fluid or non-binary, if it's mostly female stereotypes you're a "woman" and if it's mostly male you're a "man".... but all of that requires a belief in gender as an essence or soul that is present within each of us and is separate from the physical body. Some people believe we have this. Great, as long as they aren't imposing that belief onto others as fact, including vulnerable children and young adults, they can hold the belief that they have a gender identity of the opposite sex and live their lives happily doing so. Perhaps you do believe in it, perhaps not? This sentence could be read either way ⬇️
Different people experience it differently, and it doesn't live solely inside us.
I (and many others) don't believe that gender identity exists, so what those who believe in it can't do is force others to act as if it does e.g. going into changing rooms/loos or sports etc that align with their gender identity (where this is different from their sex).
But back to the OP, the son's partner clearly does believe in gender identity. The son too, presumably. I actually agree with the PPs above who say it will be unproductive challenging this directly. Whether that's with the son or the partner. That said, I wouldn't use the partner's preferred pronouns because that would involve me saying something I don't believe, just to keep the peace. Instead, I would avoid the topic and would avoid using any pronouns at all. If challenged I would explain that I don't believe in gender identity, but I'm being respectful of the son and partner's belief in it by not using pronouns and not debating the subject. If nothing but positive and active belief in the partner's identity is acceptable (to the son and partner) I would reluctantly, and with huge sadness, grieve the loss of the relationship and step away. No child or parent should ever use emotional blackmail to force the other to adopt their belief. It should be possible to believe different things, recognise that this is OK and find ways to focus on other things that are important within the family relationship.
Edited for typos and clarity.