if google lets you create the account, it's only because they were able to link it to your identity
yes, they're linking it to the identity of the school machine, or the (many other) students/teachers in the school that used that computer
If you have a unique fingerprint but they can't link it to your identity, google will lock you out of your account and delete your data because you aren't giving them $$
not sure wdym, haven't experienced that personally. just create the google account on school machine once, setup gmail's mail forwarding feature to send mails to your burner email, which forwards them to your primary email, and never bother with the google account at all. if there's something necessary you can use the school computer again.