A fourth database might be unnecessary. You can run all the themes WordPress multi-site. Each site would have it's own user profile but you could set it up so that there was a single sign on - if that is what you were trying to achieve.
Our demo sites use WordPress multi-site. You will notice that the user, appthemesdemo, is the same on all sites. We do the same thing on many other sites or systems for the same reason: single sign for all sites.
Now, if you wanted a single database with all the data for a user from all the sites, including listings, preferences, favorites, payments and more, you might need to consider a fourth database.
Cheers