I've never really understood this argument. If you can name your pet whatever you want, how does the name of other people's pets affect that? If you want yours to be unique, you still have the option to come up with a less common name. If you choose a name like "Kitty" or "Shadow", your pet never had a unique name, because those words have been used to name pets for ages. In fact, you can be more unique if there are non-unique names, because then you can have a pet named something like "Shadowly" and people are unlikely to take that name because they've all picked "Shadow" instead.
Think about it, I doubt no one on this forum has a unique name. But does that actually matter? No. We are all unique people. A name is simply used to make it easier to describe people in conversations or call out to them. If your name is Tom and there is another Tom in your class, do you think anything of it, besides, "Hey, that's cool. We have the same name."?
I actually find it strange to see this mentality of wanting unique names spring up from sites like Neopets and others that follow suit. Didn't anyone realize what a nightmare that became on Neopets? I don't want to have to name my pet something like "Mezerdia" because all the typical names are taken and then some. Not everyone wants a fantasy-sounding name for their pet or a name that reuses letters like "Jekkylll" because that was the only fitting name for the person's pet and now they have to go around all the taken ones just so it sounds and looks similar to what they actually wanted.
Can someone really tell me what the fun in claiming names is? Do you really feel like that gives your character/pet more meaning? Why does what other people have matter so much? Or is it the value unique names have on these games? The fact that you can trade them and look "rich"? Wouldn't you rather your account have value from something other than stumbling upon an untaken name? Shouldn't the value of your account be built by actual playable features on the site?
I really don't get the draw of unique names and "low IDs". Sometimes I wonder if it's just a goal to have on a site that lacks other goals for users.