1,000s of different databases simultaneously

You may benefit from disabling connection pooling (https://mysqlconnector.net/connection-options/#Pooling). By default, each unique connection string creates an implicit connection pool, which uses some amount of resources (memory, sockets). If you're connecting to 1,000s of servers, it may be best to avoid this overhead by adding Pooling = False; to each connection string. (The only downside is that connection.Close(); connection.Open(); will close then recreate a TCP (or SSL) connection to each server instead of reusing a pooled connection. But overall it may be better this way.)

What are the errors (exceptions) that I may encounter and potential strategies for managing these?

In general, MySqlException will be thrown for anything that goes wrong. MySqlErrorCode.UnableToConnectToHost might be the most common error code you see. This can be retried with some kind of exponential backoff policy. I would recommend https://github.com/App-vNext/Polly#polly.

Are there any practical limits to the number of simultaneous connections to different databases that MySqlConnector will support comfortably?

None that I know of, beyond the limitations of your OS (e.g., maximum number of client sockets that can be open). Let me know if you find any!