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).
3All pets will collect gold. I believe you have to contact blizzard support (they were quite friendly) for such purchase issues, like If pet doesn't appears in inventory.– user135338Sep 18, 2019 at 7:04
1@Sinatr: Yes, that appears to be the case. The pet appears in the wardrobe.– user598527Sep 18, 2019 at 7:12
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.
1+1 No cosmetics have any class requirement or restrictions, except for armor and weapons transmog where some type of equipment (like mighty belts for barbarians) is limited to some classes. It is true for both PC and consoles.– ZomaSep 18, 2019 at 7:25