The site you're talking to can still read your data, but most third party sites can be cut off. Privacy Badger will let you block Google Tag Manager, and while it warns you that some sites will break, few do.
I don't think there's a universal answer there, it would depend on how accurately they can fingerprint you without GTM. Blocking it does remove an identifier that would make it easier, but blocking it is also a piece of data that could feed into the fingerprinting algorithm.
It would be interesting to purposely feed a bogus GTM cookie though. It might actually throw their tracking and fingerprinting off if somehow you were able to send random GTM tags on every request.
Its not clear to me how much this will help; but based on how tags work, it seems like it should help at least somewhat. I use Privacy Badger on both Firefox on PC and Android and haven't run into any sites that break, other than maybe something like Ticketmaster? I'm sure it makes less of a difference on an Android device, where Google has other hooks to track me, but any little bit helps.
The site you're talking to can still read your data, but most third party sites can be cut off. Privacy Badger will let you block Google Tag Manager, and while it warns you that some sites will break, few do.