If it suddenly became illegal to own pets, domesticated ones, then we'd be doing a disservice for those animals. They're at the point where most wouldn't be able to survive long on their own. Perhaps over time the strongest would learn & teach their offspring... but let's face it, at the rate humanity is growing and constantly destroying habitats (displacing countless non-domesticated species as it is) there just isn't room for the population of domesticated animals to be released back into what little nature remains. I suppose there's also the chance they would be an invasive species and completely throw the ecosystem out of whack for that area.
Personally I don't see pets as just something to be owned. My wife and I own multiple cats, and to us they're practically our fury little babies. They intelligent beings. They know when we're happy and sad and respond accordingly. Heck, when we had to put down my wife's 18-year-old cat the others grieved alongside us. When either of us cry (which... uh... I don't... ever... but if I did *shifty eyes*) one/several are try to comfort us in their own ways.
They learn and teach the other as we would at school. Unfortunately this has lead to not one but now three cats able to open the knob-type doors to rooms they're not supposed to get in.
They're truly a part of the family. Not just a "pet" or "object" or anything like that.
The only time I really have a problem with the idea of "pets" is when it's something like "oh hey it's Easter let's adopt a bunny" and then suddenly those same bunnies are on Craigslist looking for a new home and going to ____ only knows what type of people. (Know of one guy who would collect them, eat the meat & save the fur for whatever.) Also exotic/non-domesticated animals - they should remain in the wild, on nature preserves, etc.