The app would look up in both databases. If it exists in any, there would be a session.
Thisnis strictly different from partitioning which I think you are mixing it up with.
Paritioning is for performance not HA
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.
There is a whole slew of downstream things you need to take into consideration.