I was just transferring a pet to someone and don't remember seeing this note before, and I was wondering what the deal with it is? Is this something to discourage/prevent people handing battle pets around for different events or something?
Yes, that a part of it. The loot and achievements are important factors.
back in the olden days people would pass their high stat pets around so their friends could gain the loot and achievements. This is also the reason pets lose some of their stats when they are transferred to another account.