The comparison was made with "gay", and whether that's a choice.
The difference is that for "gay", it doesn't matter whether it is or not.
We dealt with the gay rights movement by establishing universal rights. Anyone can get married to anyone regardless of sex. You cannot be discriminated against for the sex of your partner.
No-one gets special privileges by calling themselves "gay", so there's no problem. People can choose to call themselves gay, despite not being, and... nothing happens.
And everyone being able to have a same-sex partner despite not being oriented that way causes no more problem than everyone being able to have an opposite-sex partner despite not being oriented that way. This universal rights expansion causes no direct issues.
For "trans" it becomes an issue because people think they need special rights for "trans" people. Which is going to make people choose to identify as it. "Trans people should be allowed into opposite-sex spaces", apparently. The whole thing falls apart, because you're attaching meaning to the label, so obviously people will choose to apply the label, as it gives them something. Lots of straight men want to get into female spaces. (Is that a "choice"?)
If you just stated what you want in form of a universal right - "anyone should be able to enter any space regardless of their sex, but we should retain the male and female labelling of spaces" - then it would make sense, and would avoid the need to debate what "trans" is, or if it's a choice. Still a bad policy proposal though.