Would The Jokers trait cause a possessed character not be able to use the powers granted from the possessing character or are items and possessing different?

Yes. Anything that is equipped to a character, be it a character, object and whatnot, is referred to as an "Equipped Item". So The Joker's trait would shut make a character unable to use the effects of a possessing character.

Do Relics count as "equipped" these days?

No. Only things that have the "Equip-Effect" format are equipped.

quoted for truth.