Exact same issue. 128GB 2nd Gen AppleTV 4k with ethernet (connected via ethernet, but I've also tried on WiFi). iPhone 13 pro max. Both on the latest builds of iOS / tvOS. I am the default user and everything else seems to work (photos, airpods, game center, apple TV+, iphone remote, etc).
My daughter also has a profile and she shows up in the Facetime app, but I do not. I scan the QR code and it says connecting but nothing happens. When I select my daughter, she gets a prompt on her phone and clicks accept, but then nothing happens. I've deleted myself as the default user and re-added my account again, I've restarted both the phone and the apple tv, etc... Only thing I have not tried is a factory reset of the apple tv and/or phone, but I don't really want to have to sign into all my streaming accounts again.
@mystrizo, I'm curious what sort of network equipment you are using (I'm using ubiquiti unifi stuff).