I would use this pet (acquired via Rise of the Necromancer DLC purchase) with every character since it automatically collects gold. The pet hasn't appeared in the inventories of my other characters, unlike the cosmetic Wings of Valor item (Reaper of Souls).
Any pet that is added to your account is available for any character to Equip, using the Collection screen. You can access this by visiting the Wardrobe in camp at the Enchanter's station. You will be able to see and equip all pets available to you.
Alternatively, you can get to this screen by hitting F1 or clicking on the small mirror icon in the lower right.
I had to double check, as I've never equipped the Half-Formed Golem, but it was available to equip on my Barbarian.