I think it’s a facility provided for by X.org or (in the case of wayland) by the various DEs.
Which is waaaaay slower than using the compose key (because of mouse usage). Ever since configuring it, I found I’m using the compose key more than such menus.
Well, in quite some case – for example when creating/editing a forum post – Latex is simply not practical. Besides, Luatex, which seems to be the standard these days, supports unicode. So, why type \degree when you can get the symbol through <compose>+o+o, which saves key presses?