For the difference between major and minor keys, that's true, and my comment didn't emphasise the distinction because of the context of the parent.

But within major keys or minor keys, I don't think this is true for most people. Most people do not have absolute (aka "perfect") pitch, so if you play them a piece of unfamiliar music in a particular key, allow them enough time to forget the pitch at which it was played, then play it again transposed up a semitone, they won't actually be able to tell it's played at a higher pitch. The brain hears the relation between pitches, not the absolute pitches, and the relations are all the same.

There are certain exceptions: string instruments sound slightly different when playing the notes of one of the open strings as those strings vibrate sympathetically. Instrument bodies don't respond exactly the same to two adjacent pitches, so it's not just the pitch that changes; so does the timbre. But these are subtle effects, especially on an instrument like a piano, and they can disappear entirely on electric or digital instruments leaving only the pitches as the variation.

And there are certain limits: if I play you a piece you know very well, you'll have an expectation as to the pitch you hear it at and will be able to tell more accurately when it is played differently. If I play it to you very soon after you've just heard it, it will be the same. And even if neither applies, if I play it transposed up an octave, you'd have to be tone deaf to not hear the difference.

Yeah, while the individual 8 tones are the same. Once you start chording, A minor still sounds different from C major. On the grasping hand, there are some chords that are effectively shared so a complex piece can "disguise" one as the other.