1– It’s not part of anything = Ça ne fait partie de rien

In this example, the anything describe what he is not part of so “de rien” is a very valid option. “de quelque chose” is also accepted but could depend on the sentences.

2– He’s not a member of any group = Il n’est membre d’aucun groupe

In this example the “d’aucun groupe” describe the fact that he is not part of any group at all. The “d’un groupe” means more something like: he is not part of a group. Which is somewhat different.

Personnaly I would have said :

Ce ne fait partie de rien

Because the double negation of “Ce ne fait pas partie de rien” makes it akward imo.

Ça ne fait partie de rien

even sounds more natural…

Verboslier :

“Cela ne fait pas partie de quoi que ce soit”