Will a purchase from the Windows App Store version of Adobe software include a serial number?
No, the Windows App Store handles licensing.
So there is no serial number or product code. You DO need to download from Microsoft, definitely do NOT download from Adobe, you won't get the version that is licensed for the Windows App Store. If you did this, completely uninstall it, but even that may not be enough.
I lost a lot of time understanding the difference between Adobe PhotoShop Elements from Microsoft Store wich have no serial number and Adobe version wich have one. Not clear and not an easy way to sell a software.