It is possible to have plugins use their own DB and have things call out to that to retrieve info - e.g. CiviCRM uses it's own database to store CRM info (although it's also MySQL) and can provide forms for Wordpress pages etc.
You can also do some pretty nice data syncing/access between the two using Postgres' Foreign Data Wrappers to access MySQL data within Postgres - unfortunately it doesn't look like MySQL has a equivalent for accessing Postgres data from within MySQL.
You can also do some pretty nice data syncing/access between the two using Postgres' Foreign Data Wrappers to access MySQL data within Postgres - unfortunately it doesn't look like MySQL has a equivalent for accessing Postgres data from within MySQL.