I don't understand how "cat" is a gender though?
Of course, it isn't; but I think the main issue is that 'gender' has crept and become a byword and a passport for claiming identity 'rights', with hardly anybody stopping to ask why this is the case. Linguistically, it's a bit like the 'oholism' on the end of 'alcoholism', where the suffix is clearly just 'ism', but lazy people decided to add on most of the word 'alcohol' and then nonsensically apply it to make words like 'chocoholism' and 'workoholism'.
It was originally (many would say deliberately) merged to be used as a synonym for 'sex', with it paradoxically retaining its own very subjective flexibility of social nuance, yet also sailing on the coat-tails of the objectively defined 'sex' when it came to declaring it as established undisputable fact.
Once enough activists gained enough ground in redefining the word 'gender' and mangling it into a powerful force for propaganda, they'd have been stupid to try to re-invent the wheel and not use it as a verbal talisman with which to beat people with associated assertions that 'this is fact' (because it's been associated with the newspeak definition of 'gender'), even when defending the most absurd concepts imaginable - many of which have no relationship with sex or sexuality at all.
Saying "I identify as/think I'm a cat" will rightly draw derision and/or offers of mental health treatment; but if you cloak it in that magic suffix and say "I am catgender" then it becomes something that everybody is gaslighted into simply accepting and nobody is allowed to question it - or the brave ones who nevertheless do question it can be safely dismissed as somethingorotherphobes, which makes them instantly the bad guy.
The pronouns we use for an individual are based on their gender and although we may assume a person’s gender based on their name, appearance, voice, or expression, and assume which pronouns to use for them, this may not always be correct.
I've been a native English speaker for over 40 years and I've never used sex-based pronouns to relate to anybody's arbitrary idea of gender - and I don't believe I'm rare in this. For centuries, somebody would introduce their new baby boy or girl and (nice) people would respond with "Oh, HE [or] SHE is beautiful, based solely on the baby's sex."
If it were true that the pronouns we use are based on gender - a fluid social identity and self-expression system (often based around sexuality, attraction and other mature impetuses) - and not biological sex: male or female, everybody would refer to a known, born, individual baby as 'they' for many years until they could have a hope of knowing what that individual's 'preferred gender identity' had been decided as.