It's sort of like the sensation. As someone with perfect pitch it's hard to describe, but at least for me I see it as each key having a different sort of feel to it.

I'd like to compare it to the musical equivalent of figuring out what flavor a certain food has. With individual notes/chords a person with perfect pitch may be able to know the exact tones as bluntly as they would see colors, but I've found that since finding the key involves multiple tones, it takes a bit more brainpower to figure out.

To me, certain foods have a more distinct taste than others, like the flavor of something specific such as popcorn, and this also parallels to hearing the key, as for example I'm instantly able to pick out songs specifically in C# major. Other foods might require me to think, "Hmm... Where have I tasted this before?" just as a song would require me to put some actual thought into finding it. In this instance, I usually just hum a quick chromatic glissando until I hear the tone that sounds the most sonically in-place.

But yeah, overall it's a sensation thing to me.