There isn’t necessarily a correct architectural answer here. It depends on some of the constraints of the system.
I have an option for using map[string]*sql.DB where the key is the url of the database, but it can be hardly scaled when we have numerous number of databases.
Whether this will scale sufficiently depends on the expectation of how numerous the databases will be. If there are expected to be tens or hundreds of concurrent users in the near future, is probably sufficient. Often a good next step after using a map is to transition over to a more full featured cache (for example https://github.com/dgraph-io/ristretto).
A factor in the decision of whether to use a map or cache is how you imagine the lifecycle of a database connection. Once a connection is opened, can that connection remain opened for the remainder of the lifetime of the process or do connections need to be closed after minutes of no use to free up resources.
Should we have a sharding layer for each incoming request sharded by connection url, so each machine will contain just the right amount of database connections in the form of map[string]*sql.DB?
The right answer here depends on how many processing nodes are expected and whether there will be gain additional benefits from routing requests to specific machines. For example, row-level caching and isolating users from each other’s requests is an advantage that would be gained by sharing users across the pool. But a disadvantage is that you might end up with “hot” nodes because a single user might generate a majority of the traffic.
Usually, a good strategy for situations like this is to be really explicit about the constraints of the problem. A rule of thumb was coined by Jeff Dean for situations like this:
Ensure your design works if scale changes by 10X or 20X but the right solution for X [is] often not optimal for 100X
https://static.googleusercontent.com/media/research.google.com/en//people/jeff/stanford-295-talk.pdf
So, if in the near future, the system needs to support tens of concurrent users. The simplest that will support tens to hundreds of concurrent users (probably a map or cache with no user sharding is sufficient). That design will have to change before the system can support thousands of concurrent users. Scaling a system is often a good problem to have because it usually indicates a successful project.