This may be related to the issue we are experiencing:
https://wicnews.com/science-tech/microsoft-shutting-cortana-app-ios-android-014023442/
https://support.skype.com/en/faq/FA12091/why-are-some-features-not-available-in-skype-anymore
That info isn't related but I joined your thread since the Cortana Skype Linking is the main issue in your thread. Here's the link for everyone here:
Now that still doesn't help me on the Xbox Commands specific to the Invoke at all. Can anyone else confirm that the shortened Commands "Hey Cortana, Xbox On" are not working for them anymore. Cortana just says "Something went wrong please try again at another time" but at least she isn't doing a search for "Xbox On" and confusing it for Xbox One and then telling me about the Xbox Series X anyway (I have never seen a smart speaker be so stupid).
While the long commands work "Hey Cortana, tell Xbox to turn on" it's more of a mouthful and since you have to say more, there's a higher chance she'll misunderstand you. Launching games or apps is more difficult with the long commands too. I do know I have it paired with my Xbox as when I say "Hey Cortana, Launch Netflix" she again says "Something went wrong please try again at another time" rather than saying that she can't launch apps from here (this is when the Invoke isn't paired to any Xbox One, so she tries to open Netflix on the Invoke to which she can't obviously).