Implanting RFIDs in pets is extremely common, as a method for identifying them to return them to their owners if they are lost or injured[1]. I agree it would a little silly to get an implant strictly to open a pet door, but given that they are already often present, leveraging that to open the door seems like a pragmatic choice.

1: https://en.wikipedia.org/wiki/Microchip_implant_(animal)#Use...