From the new AMA:
"The vanity pets in ESO are accessible account-wide. Any character can read the in-game mail that the vanity pet is delivered in, and retrieve the attachment. When a character retrieves an attachment, the attachment is removed from the in-game mail and placed in the character’s inventory for use. If you wish to use the vanity pet on a different character, you can simply place it in the bank and retrieve it with the new character. Each attachment is a single item."
This makes it seem like you simply get a single vanity pet in the mail, and if you make a new character you do not get another vanity pet for that character.
This caused some huge issues with Guild Wars 2 launch becuase many people did not realize this and deleted a character that had it in their inventory, etc.
Also for people that play more then one character (IE alts) this means they will constantly have to shuffle the pets around their active characters back and fourth, which is annoying.
Why not simply make the pet get sent in the mail to each character that is made on that account? This is MUCh easier on the players and solves the issues of accidentally deleting it when it's on a character.