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It is possible to use Game Pass for PC in a game-sharing setup, but all involved PC accounts must be signed in to the same account via the Windows Store. I reached out to Microsoft Gamer Support, who confirmed that you can not be logged in to both PCs at the same time.
Can I use my Xbox Live Pass Ultimate Subscription on multiple Windows 10 devices?
Yes, just like Xbox Live Gold or Xbox Game Pass, you can use this subscription wherever you sign in. Your subscription is tied to you, not to a specific device.
- Xbox Game Pass Ultimate FAQ
"You can only do so one at a time. It will log you off when logging on a different pc."
- Microsoft Gamer Support
Ultimately, you have three options:
Log in to your account via the Microsoft Store on your son's PC, and arrange a schedule so you are not both trying to play Game Pass games at the same time.
Purchase your son his own Game Pass for PC subscription
Forgo Game Pass for PC all together, on your son's PC, and hope that they release some sort of sharing bundle, in the future.
The different Xbox applications shouldn't make any difference; to clarify, both applications are the same. You have the beta version, which is an older pre-release version; your son has the newer version, downloaded directly from the Windows Store.