The MySQL (not PostgreSQL) documentation mentions that instances have a static IPv4 address automatically assigned. But does the same apply to PostgreSQL and is a static IP address also permanent?
That is correct.
PostgreSQL instances exist only as 2nd gen flavor, which means there is no support for IPv6 connectivity for it yet.
Instance IP address remains associated with it throughout its entire lifetime. (i.e. restarts, maintenance, failover, etc. won't cause a change of instance IP address.)