No it doesn't. What gave you that idea? You still have all "ACID guarantees" within each database.
>You can’t do cross-database transactions (to my knowledge),
That's true of most databases. If you have two apps and they use two different databases you won't have transactions across those two apps.
>which means you can end up with corrupted data during aggregations.
No, aggregations within each database work as you would expect.
>What am I missing?
As others have said, you use this pattern only if you don't intend to cross databases. By the way, in NoSQL databases like MongoDB, every document is its own little database so having a per user database is a massive upgrade over NoSQL.