Database for Python Twisted
Scalability
twisted.enterprise.adbapi isn't necessarily an interface for talking to databases in a scalable way. Scalability is a problem you get to solve separately. The only thing twisted.enterprise.adbapi really claims to do is let you use DB-API 2.0 modules without the blocking that normally implies.
Postgres
Yes. This is the correct answer. I don't think all of the Python bindings are abandonware - psycopg2, for example, seems to be actively maintained. In fact, they just added some new bindings for async access which Twisted might eventually offer an interface.
SQLite3 is pretty cool too. You might want to make it possible to use either Postgres or SQLite3 in your app; your unit tests will definitely be happier running against SQLite3, for example, even if you want to deploy against Postgres.
Other?
It's hard to know if another database entirely (something non-relational, perhaps) would fit your application better than Postgres. That depends a lot on the specific data you're going to be storing and the queries you need to run against it. If there are interesting relationships in your database, Postgres does seem like a pretty good answer. If all your queries look like "SELECT foo, bar FROM baz" though, there might be a simpler, higher performance option.
There is the txpostgres library which is a drop in replacement for twisted.enterprise.dbapi, —instead of a thread pool and blocking DB IO, it is fully asynchronous, leveraging the built in async capabilities of psycopg2.
We are using it in production in a big corporation and it's been serving us very well so far. Also, it's actively developed—a bug we reported recently was solved very quickly.
You could look at nosql databases like mongodb or couchdb with twisted.Scaling out could be rather easier with nosql based databases than with mysql or postgres.