I spoke with Tim Starling at WMF Tech 2018 and he recommended that I removed Gamepedia's/Hydra's hacks for Aurora by moving them to a MySQL driver. The only compatibility difference between regular MySQL and Aurora is how it handles getting replication status. The original code has been tested on gamepedia.com for about two months now. This code differs in that it has been split out into its own file. Currently it is being tested in the staging environment. Example wiki: https://meeseeks.gamepedia.io/Meeseeks_Wiki
MediaWiki currently has two db implementations that WMF are volunteer-maintained and not tested in WMF CI (Oracle, Mssql). I do not know if we want to expand that, or whether the idea is to make this possible for an extension to provide.
From talking with Aaron, I believe it is already supported for an extension to provide an IDatabase implementation, and to be able to use it through MediaWiki configuration. However, I don't know if this has been tried before.
After some discussion in CPT, it seems like this should be done as an extension. Adding more databases to core is probably a bad idea, considering we have a bad track record of maintaining support for the ones we already have. Doing this via an extension is a bit of an experiment - the general case is blocked on T191231: RFC: Abstract schemas and schema changes, but Aurora should be 100% compatible with MySql, so that shouldn't be a problem.
However, we'd have to make sure this wouldn't fail due to hard coded checks for "mysql" or "mysqli" in core. Any such checks would need to be replace with instanceof checks against DatabaseMySQL.