Louella:
Yes. Yes, it would.
However, I'm pretty sure that scenario makes New Eden's cyber-ethicists twitch; I'm not at all sure that the legal bar on uploading people is actually limited to people.
Entertaining thought, though. Oh-- wait, an actual issue.
Smell. A dog's identity is highly keyed into the sense of smell, right? Hence all the butt-sniffing upon first meeting. Now, I certainly don't know this for sure, but it seems like you'd need to have some way of making sure that any given copy still smells like itself to itself, and that any other cyberpooch (including other uploaded copies) smells different. I was going to suggest that you could gradually change the smell of a given copy, but that seems troublesome.
What you might be able to do is include some sort of algorithm that makes any given copy always smell, to itself, like the original, and makes any copy with a different serial number smell different-- possibly through a scent-modification based on the serial number. Hm.
Still illegal, probably, but a fun idea to play with.