Black Hole by default is Cold damage, shows it as a scaling tag.
Black Hole with the Red Giant turns it to Fire damage, showing Fire as its scaling tag.
Further going down that path, Binary System, makes it deal both Cold and Fire damage. It is correctly increased, as when equipping an item that only increases Cold and Spell Cold damage, it dramatically increases the damage (tested with Frozen Ire), however, the Scaling Tags does not show Cold there as it should.
Edit: To verify using a better method I just flat out equipped and unequipped an Idol with Cold Damage Over Time. It did raise and lower it, so it’s working properly (verified) just needs the tag.