Like @anon63664082 said you’ll both need the latest cross-play version installed. Furthermore I did experience the same issue with my son who’s on Steam while I’m on the MS Store client.
I did managed to solve it by hosting public. Please make sure your friend also changes the region filter to all regions in MP lobby search screen.
It looks like the game invites don’t come through in the Xbox Companion-console or in-game at his side at all although we are friends through our MS accounts. Will test this more and get back on it later.