IGDB, authentication and tokens, need server app?

TLDR: Yes

Longer answer: You need a Proxy server because you need to generate an App Access Token.
IGDB only accepts app access tokens, and the only way to generate one is using your Client Secret.

So the client secret cannot go in the Mobile app as that as violates the developer agreement
The app access token cannot go in the Mobile app as that as violates the developer agreement

Both of these are essentially passwords and should be treated as such.

A Proxy server is a good idea, as your Proxy would also cache data from IGDB, so if someone looks up details about the game Halo: Infinite and then a second person does, you can load from you cache instead of calling the API which makes you respond to the user more quickly. Which is even better for mobile users as user won’t have to wait as long

Different API’s have different rules.