Maybe I have something different in mind when I say "status". To the extent that an agent's treatment of an opponent is based on some sort of persistent state, rather than on the opponent's local behavior, I would say that that state is a form of status information. (Thus, one can have a status with a grim trigger, but not with a tit-for-tat.)

In humans (and I conjecture in approximately-winful strategies generally?), status tends to be one-dimensional, with one end eliciting "do nice things for this person" behavior, and the other end eliciting "do mean things to this person" behavior.