> The app would look up in both databases. If it exists in any, there would be a session.

And if you find the session with differing values in both databases, how do you know which one is up-to-date?

You need an algorithm to pick which data is right, such as electing a master instance.

And that brings us back to the original discussion: to manage sessions (unlike caches) in a highly available way, you need to setup HA (or reimplement it, which obviously is a bad idea). You can't read round robin from multiple non-HA instances.