At the extreme end: If my Javascript frontend is being told about a database configuration error happening in the backend when a call with specific parameters is made - that is a SERIOUS security problem.

Errors are massaged for the reader - a database access library will know that a DNS error occurred and that is (the first step for debugging) why it cannot connect to the specified datastore. The service layer caller does not need to know that there is a DNS error, it just needs to know that the specified datastore is uncontactable (and then it can move on to the approriate resilience strategy, retry that same datastore, fallback to a different datastore, or tell the API that it cannot complete the call at all).

The caller can then decide what to do (typically say "Well, I tried, but nothing's happening, have yourself a merry 500)

It makes no sense for the Service level to know the details of why the database access layer could not connect, no more than it makes any sense for the database access layer to know why there is a DNS configuration error - the database access just needs to log the reasons (for humans to investigate), and tell the caller (the service layer) that it could not do the task it was asked to do.

If the service layer is told that the database access layer encountered a DNS problem, what is it going to do?

Nothing, the best it can do is log (tell the humans monitoring it) that a DB access call (to a specific DB service layer) failed, and try something else, which is a generic strategy, one that applies to a host of errors that the database call could return.