Stranger Things Theory:The school therapist isn't a real therapist.
So I saw a video by a Youtuber called Georgia Dow, who is a therapist and analyses a lot of fictional characters including ones from Stranger Things. On one video she analyzed the session between Max and the therapist and even she noted that the therapist's practices are a bit odd. If you notice she constantly pushes Max for information for her living situation and her trauma. According to this youtuber a real psychologist would never push a patient for information and would let them speak naturally. This shows that she just wants information from Max and not to really help her. Also not to mention that she constantly tells Max that she knows she is lying to her, which again a real psychologist wouldn't do that. They would let you feel comfortable and not pressured.
The second part of this theory, revolves with her involvement with Vecna. If you notice all of Vecnas victims had first gone to this therapist(with some minor exceptions) like Fred, Crissy and Max. The therapist actively kept files on them(which is normal), but if you notice the contents of this files were later used by Vecna. For example she knew from Max all about her mum and of course about Billy he also knew about Fred's car accident and Crissy's issues. All of this are secret things that you would only share with a therapist.
Now let's get on to the why.Now I have a small theory about this. I think she is possesed by something simillar to the mindflyer but this time Vecna controlles it. If you notice he is covered with vines and flesh from the upside down so it wouldn't be hard to believe that he is one with it at this point.For this reason it wouldn't be hard to believe that he could puppeteer the Therapist. Also my friend mentioned that she has a key on her neck and which may have something to do with a keyhole on the grandfather clock(personally I haven't noticed it, so take it with a grain of salt). Lastly she kinda acts like Billy did when he was infected. She is constantly cheerful acting as natural as possible and she gets annoyed when she doesn't get her information.