You can remain anonymous to the people in those groups. No VPN required either, just a different Facebook account.
You can’t be anonymous towards Facebook itself, because that social network requires an account.
You can’t be unidentifiable by Facebook, because they would require you to very your account.
You can remain unlinked to your real identity (as seen by Facebook) if you use a different SIM. And a different phone if you’re planning to use the app.
You can’t remain unlinked to your real identity by the government because you need to provide your id when buying a new SIM, so when necessary they can query Facebook for your phone number and then query your phone provider for your id.