Change the SQL Database Name of a Data Source based on User ID

You might be able to do this if you used direct query and looked at this post by GilbertQ - Power Query (M) – Passing Parameters dynamically to a SQL Server Query - Data Analytics | FourMoo and Power BI

But not sure how performance would be, and what things Direct Query will mess up for you - you cannot use Power BI Time Intelligence.

What I would be more inclined to do is create a view on the server that combines all relevant fact tables into one, with a new field that indicates the user name, then use RLS to filter the data at the report level.