When a Rails app starts in production mode, it preloads its classes and retrieves e.g. column data from the configured database. To use multiple databases, you would have to stop it doing this - and that could potentially cause a lot more problems than it solves.
A much easier to manage solution is simply to have two copies of the app, one listening on each domain.
A Rails app is generally designed to run backed by a single database, if you have data that is only specific to a single domain then your data should maybe all be in the same database, but have the relevant models linked to a particular "SiteDomain" model which you can then content-manage.