I need to sync my phone with my daughters icloud

I don’t use Facebook or Facebook messenger but what happened is that while you were sharing an iCloud account, you also shared all of that content so it is associated with your daughter’s iCloud account. Once you decided to have separate Apple ID’s, whoever creates the new ID loses access to the data associated with the original ID.

I assume that you will have to create your own Facebook account now. There are ways to download data from another iCloud account and add it to a new iCloud account, but I don’t know how Facebook works at all.

Apple ID's aren't meant to be shared.

Who told you that? It makes no sense at all.