You could never transfer mounts. What they did offer is a manual process where they would send you a new mount if you show a proof that you deleted your old one. All you mount coloring would be gone, because you had a new mount.
By that token, any item shop is transferable, not just mounts.
Close, but not quite in the black.
FA used to transfer permanent mounts from one character to another, including across accounts which were yours. Only permanent mounts which had been purchased and not gotten through a promotion, a gasha, or from an event could be transferred. To the best of my knowledge, FEU never did mount transfers.
The process was as you described, for the most part. It required 1.) asking to transfer the mount from character X to character Y, 2.) Support acknowledging the request, asking for the mount to be deleted from character X and sending response to Support, 3.) Support would make sure the mount had been deleted, would then spawn replacement mount for character Y. It took up to a week to accomplish the transfer. Spawned mount would be "normal", with no custom coloring.
No other Item Shop product could be transferred from one account to another.