“tell me if the queen really has to appoint him, or if its just for show.”
Its a bit of both.
The Monarch has more of a veto over the ruling Party’s pick, rather than an ability to appoint who they want, but there is some constitutional power.
Ceremonially though, the appointment meeting of a new PM with the monarch, is kind of like a show of blessing/annointment.
So then the monarchy still controls uk government.