She wanted a place to belong because she kept getting vilified to her face even though her dad didn't realise it was her he was vilifying. She couldn't open up about being Spider-Woman to her dad because of his whole demeanour about it.
Just like trans people just want somewhere to feel safe whilst they're getting vilified to their faces by their so-called "family" and feeling like they can't open up to them about it.
And then you throw in the "Protect Trans Kids" flag on her wall, her dad wearing a trans flag on his uniform and the use of the trans flag colours when she comes out and talks to her dad about being Spider-Woman and wanting him to accept her.
It's the use of flags and colours that make this an allegory and Spider-Verse is extremely rich in using colours to mean certain things.