Is it possible to play Minecraft on a separate game server with others, if the individual players have different consoles (eg Xbox and PS3)?
This is called cross-play and yes, you can in Minecraft, see https://techcrunch.com/2017/06/11/minecraft-is-now-available-for-cross-play-on-any-device/
On this site where the "Better Together Update" is described, but unfortunately nothing of the Playstation, but "only" Windows 10, iOS, Android, Xbox One, Nintendo