At present, you can have different Role and Department text in each organisation (I haven’t checked workspaces) by editing your profile while in that organisation. Two things occur to me:
It would be good to be able to set a profile photo per organisation/workspace, not just per account.
The email tabs of the profile configuration work across all organisations, and give the options for all organisations, whenever they’re open. If parts of the profile are set per-organisation, it would be helpful to make this explicit in the user interface, rather than relying on the user to discover that they can change these in each organisation separately. Perhaps a dropdown at the top of the profile to say which you’re currently editing, with any fields that don’t change between organisations above this?