You can actually have your database hosted a couple of different ways with Lightsail.
Lightsail has a managed database option that is built on top of RDS, so if you don't want to have to install and manage MySQL you can just use the managed offering. It will likely be more expensive than self-hosting, but comes with the benefit of not having to manage the underlying system, optional one-click high availability mode, and automatic backups. Today the managed service only supports MySQL - but we've announced that we're adding Postgres shortly.
If you'd rather pay a little less, you can (as stated above) install just about anything you want into a Lightsail instance. So you could provision a Lightsail instance and install MySQL (or Postgres or Mariadb or whatever).
Now if you want to run MSSQL, there isn't a managed offering, but there is a blue print for SQL express on Windows server available as well.