AI don't have consciousness, but they do possess something related to consciousness while still not being quite there, what is this thing that both humans and some AI both have presently?
The problem here is that there is not general agreement as to precisely what consciousness means. It is obviously the state of being conscious, and when applied to a person it means that they are not unconscious (confusingly the medical definitions seem to work that way around, though I'm not a medical expert so I might be wrong about that). However, when considered as an abstract characteristic of a being, the definition is... slippery.
Other words that might be used include sapience, literally meaning wisdom but often also used for this slippery idea that you are alluding to. Then there's self-awareness, the idea that a being is, well, aware of itself. Cognisance would be another, which is an old word meaning knowledge or awareness, but now applied to capacity for such knowledge or awareness. Reason refers to the capability to think and reason deliberately. I expect there are more, but those are off the top of my head.
These words might be used in science fiction and fantasy to describe the characteristic that unites both human and non-human "races", and can also be used both in fiction and non-fiction to talk about thinking machines. I've seen several used in popular writing about the sort of AI that is currently in use, and what might be developed. Because definitions are not consistent, however, writers will disagree as to what terms apply to what entities.