The Wikipedia article on AGP slots does indicate that swapping cards between computers with different chipsets may find "incompatibilities due to firmware." The article isn't clear whether that's computer firmware or card firmware.
The AGP standards were pretty good, but there are apparently a number of "flavors" that various manufacturers used that aren't well documented.
AGP appeared during the fairly early Win98 days, but it's extremely rare now to find a new PC offered by "consumer market" makers that has AGP slots. Virtually all of the "standard" machines offered are using PCI for graphics cards, and my local retail shops don't appear to be aware that AGP cards are still available. (And most new "assembly line" computers don't have enough PCI slots to allow much in the way of upgrading - or even maintenance. They may have several slots, but they ship them ASF = All Slots Full.)