I don't know about Bernalillo County but this has been a City of Albuquerque requirement for many years. It may have started with the HEART Ordinance of October 2006. An Albuquerque City Councilor, I think from the north valley, sponsored the ordinance. The ordinance limited number of pets, required microchips, etc. and required cats to be confined to your property and subject to a leash law
. I purchased the yearly license when I took my cats into the vet for their annual exam and shots. However I did not get the impression that the vets enforced it much - more like "do you want to buy a license or not?" But once you purchased the first license and got into their system, some company in Texas sent you a reminder every year.
I'm not surprised that Bernalillo County now requires a license - their website states this is the case. But I could not find when this requirement started.