I bought the Virtual Desktop app for my Quest 2 on my browser but when I open the Oculus app to install it, it's not in my library or in the store. When I search for it in the oculus browser I find it in the shop but not the app. I also checked the purchase history for the same account on both the browser and the app and the browser said I bought it but the app says I didn't, even though it somehow has my credit card information even though the only thing I've bought off of oculus is the Virtual Desktop app.
There's 4 different Oculus stores: Rift, Quest, Go and GearVR, and games bought on each go into different libraries. Virtual Desktop isn't "cross buy", so buying on one store doesn't give you access on the other stores. If you went to the Rift web store, the Rift PC app or the Rift section of the mobile app and bought Virtual Desktop, you only got the Rift PC version, not the Quest version.
If a game is cross buy (some are, some aren't), buying it on Rift gives you a free Quest version and vice versa, but they still won't appear by default on the other platform, you have to go to the other store and the buy it, but the price will be listed as Free.