[Question] Which key are modes in?

So from my understanding there’s 7 modes derived from each note in a major scale. The Aeolian mode is the relative minor to the Ionian mode. So if I’m playing in the key of A minor I’m playing the same chords and note as if I’m playing in the key of C major, but with a different tonic/root (idk the correct term).

So if minor keys are just a mode of a major key then what key are the other modes in? If I play D dorian am I playing in the key of C major/A minor or is there such a thing as D Dorian key?

I understand everything is a bit more nuanced and not as black and white, but if someone could explain that’d be awesome, thank you!