No, the problem is not the person you replied to, the problem is that some people mistakenly believe that identifying as something means that one is that thing, and that therefore, they should be treated as the thing they identify as (including having the rights afforded to this group).
This is not the case. Identifying as something is not a sufficient condition for being that thing. This boy identifying as a girl does is not sufficient for him being a girl. Further, it is not a necessary condition either. This person is a boy even though he does not identify as one.