Other than F.I.C.S., where can a friend and I analyse our games online together free where both of us can move the pieces on the board.
Lichess.org has a feature called studies. You can add your game as a chapter in a study, invite your friends as contributors and make that study public or private. You have a (rather weak) engine analysis, a small database of chess games, and a chat. You can annotate your game, add alternative lines and so on. The multi-chapter option makes that very useful for much more than game analysis. I'm using it primarily for opening studies.
Here is a screenshot from my Caro-Kann study:
Not free, but also worth a mention: Chess24.com has an analysis board for premium members. Very similar features, but there is only one game per analysis, not multiple chapters. The database is better (larger and properly cleaned up names, lichess is sometimes a bit annoying in that regard). You have to start a broadcast (see screenshot), and then you can invite your friend who doesn't need a premium membership for that. The downside is that the chat vanishes once you stop the broadcast.
In both cases you can move pieces on the board, and all other participants can see and annotate these moves.