Can a user have different roles on different productions?

Yes. A person's role is set production by production, so someone can be a Manager on one show and an Administrator on another.

To change it, go to Administration > User Administration, click the cog on the person's row, choose Edit, then Production Roles. Each production has its own VirtualCallboard Role drop-down. See Assign users to productions.

The exception is the Administrative roles, which cover things like adding users or starting productions. Those apply to the whole site and are set once, under Administrative Roles on the same screen. For how the two kinds fit together, see User Roles & Permissions.