A query for a given item goes to a router*, as you said, that directs it to a given shard which holds the data. I don't know Uber's schema, but usually the data is "denormalized" and you are not doing too many JOINs etc. Probably a caching layer in front as well.
If you think this sounds more like a job for a K/V store than a relational database, well, you'd be right; this is why e.g. Facebook moved to MyRocks. But MySQL/InnoDB does a decent job and gives you features like write guarantees, transactions, and solid replication, with low write latency and no RAFT or similar nondeterministic/geographically limited protocols.
* You can also structure your data so that the shard is encoded in the lookup key so the "routing" is handled locally. Depends on your setup