Yes, it is supposed to be bound, and no, it cannot.
"My frozen dairy-based confectionery attracts all the males of the species to the facilities. They all agree on it's superiority. Indeed, it is superior to yours. I could teach you the finer details but that would require monetary recompense on your part." -The Milkshake Song: Vulcan Edition
Not true 100% of the time. Some Boffs bind on pickup. To name a few, the Reman bridge officer and the Borg science and engineering officers. The Breen bridge officer, for some reason, probably as an oversight, can be transferred by way of mail or the exchange. I don't know which of those categories the diplomatic Boff is in, but if Tiggy can't move it, I'm guessing it's in the former group.
And many Boffs can still be transferred from person to person through direct-trade once they've been commissioned, a rather backward consequence of it needing to be possible for engineers to train their tactical and science fleetmates' bridge officers, and vice versa... Would've been safer if they'd let us "Train/dismiss" active Boffs, then people could just buy any common Doff trained in the wanted skill from any random stranger, and apply it themselves to their Boff of choice, without risking losing said Boff to said stranger.
Anyway, if you want to move your diplo-Boff from character A, to character B, you might try this. Get a friend or fleetmate you trust, use direct trade to trade your active diplo-Boff to him/her (unequip the Boff's gear first), then log into your alt and reverse the procedure. It probably won't work, but there's always a chance.