I honestly don't know how common this is, but we do have some use cases for this, yes. For example Wordpress with CiviCrm - CiviCrm is installed as a Wordpress plugin but uses its own database. This is my current reason for needing this. They may also be situations where a LAMP app is installed on a domain or subdomain, another, app is intentionally installed in a subfolder and both need a database.
Maybe I have misunderstood the way access to databases in LAMP app works though. Am I correct in saying that one app cannot access a database set up in another (LAMP) app? If that is not the case then maybe there is no necessity to have multiple databases and a LAMP app could be used simply to host the additional database?