Having a database per tenant has always interested me. It make sense if the databases will be different (contain other tables)
Adding a tenant id to the tables that require them makes sense.
The best approach is to create a pivot table between users and tenants that holds user_id and tenant_id. Use the primary id of that table to represent the user at that tenant.
A separate database is best used if you need an absolute firewall between the tenants for security reasons. Almost no bug can ever let one tenant see the data from another. If they share tables, it's easier to make that mistake.
The downside is you can't have quieres across all your tenants, so you'll want a BI database that brings all the data together anyway.
So no point of overthinking it.
Doing a little research first, and thinking about which tradeoffs matter most in your particular case could save you a lot of effort rewriting things down the road.