Or is it possible to be a philosopher-monk, for instance?
In tabletop? Certainly!
In BG-3? Most likely ... since just as with all other classes, we most likely will just get [MONK] dialogue options, but will still have all other generic options that are available for everyone.
So even if every single [MONK] option would be somehow related to some deity ... all you would need to do is not picking them.
