Reading between the lines on the recent AMA posted on the official site, it appears that there is a shared bank for all characters on an account. This came up in the discussion about the vanity pet as it seems this is a single pet that is bound to the account, not a separate one for each character on the account. The person answering questions stated that "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."
This also indicates that vanity pets take inventory space, which isn't a good thing. I won't be inclined to collect them if they're going to waste my inventory space. It's possible, too, that they could go into a "special area" that doesn't take inventory space but that you can pull the pet out of to move it to a new character.
Does anyone have more info on how bank storage works in ESO? Is it a single large shared bank for all characters, like in GW2, or is there individual character bank space and a small set of shared bank space like in LotRO? (Pretty sure it was LotRO that did that.)