https://support.xbox.com/en-US/xbox-on-windows/accessories/xbox-wireless-adapter-for-windows-setupIt looks like a Bluetooth dongle which pairs one controller to one host computer. Like the Bluetooth dongles used on wireless mice and keyboards.
ROG ZEPHYRUS GX501 specs list support for Bluetooth 4.1 ... although there's several GX501 submodels, and I'm not sure if they all come with integrated Bluetooth.
But I've used other laptops with built-in Bluetooth, they can pair with most Bluetooth devices without using the dongle (although the dongle is usually needed once, when setting up the device pair for the first time). I don't see any reason why it wouldn't work on the GX501 unless Microsoft designed this Xbox controller dongle to deliberately force hardware dependency.
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[/Korth]