You have to have one database for logins which will contain connection string to use for that particular client and redirect him further into the application. You have to take care to maintain the connection string in session, cookie or some other persistent way.
Possibly better design (depends on your particular setup, but for web it is almost always better) would be to simply have single database and qualify all relevant tables with clientID. That way data that is common to all (currencies, list of countries etc) can be common lookup tables and products and categories would be all in single table, but with clientID.
i.e.
TABLE products
product_id int
client_id int
product_code
product_name
price
discount
Then on login, you know which client is logged in and all queries will be filtered by client if it is relevant
SELECT * FROM products where client_id = @client_id
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