putting it in quotes is done usually to try to exclude or differentiate trans women from women in general. This is transphobia. Trans women are women.
>Trans women are women.
What is a woman?
The fact that this is an obvious lie makes it a better shibboleth, because people in your outgroups are hesitant to pay the reputational cost of repeating it. In a case like this one, the lie can get people killed by encouraging underestimations of a man's propensity to violence.
We do not need to play along with the absurd claim that if a man identifies as a woman that means he is a woman. This is sexist nonsense.