If there's a will to Syphon money from the tax payer, there's a way. First, a large scale armed forces are hired to do the dirty work. And then, a new department to register dogs for a fee(even there's a ton of human alien who willing to pay but still doesn't)
It is also my believe that people tend to NOT keeping house pet anymore since not many are paying for the ID. My neighbourhood, those "stray dogs" actually have their own rice bowl somewhere beside on of those house gate. I guess they just don't wanna own the dog but still wanna see a dog around.