"Elements", yes, for example, wood for liver, metal for lungs and so on. Spirit, no.
So I take it, TM uses Chinese elements (wood, fire, earth, metal, water) when discussing the organs, but Hindu elements (eart, water, fire, air, space) for everything else?
AFAIK, while there are spirits or demons associated with most diseases, called "disease lords", they are not specific to a given organ, but rather govern a class of disease as a whole.
I should clarify that by "spirits of the organs" I was referring to the wu shen (5 shen) - not spirits in the sense of entities that cause disease (which also exist in TCM), but 5 aspects of consciousness of a human being that are said to reside in the 5 solid organs.