My usual way of finding an approximate pitch is to quickly and quietly feel how it lies in my vocal range. (This would be without an instrument. With an instrument obviously I just play it.) This gets me, too, within a couple of semitones. I can do it without making any audible sound, because years of singing has connected the internal audiation of pitches with the muscle memory of how my vocal tract produces them. But not to anything like "perfect pitch" accuracy.

Will, I don't doubt you can pick out the right key on a guitar audibly, like I could find that Bagpipe A, but with guitar I wonder how much of it is hearing the chord voicings and the sound of open strings vs stopped strings? As you know E major in open position sounds considerably different from the same name chord played in the open D position with a capo on II.

This is interesting, me here talking like my ears are really good -- in these respects they are -- while at home I'm trying to learn to mix better, and finding my ears struggle to pick up the supposedly dramatic differences between examples on tutorial recordings. I suppose it's a matter of time and experience.
-Glenn