I wonder if human singer instinctively chooses another note, that is not 100% same frequency as in piano. You know, there are always imperfections in piano tuning even if it's done in today's standard way(all intervals are not perfect). I'm not a piano tuner but this is my understanding. Possibly trained singers can sing in a better harmony somewhere where piano gets it (very) slightly wrong?

It's not (just) instinctive. Good choral singers adjust their tuning purposefully to match the overtones of the harmony.

For example, if you are on the fifth in the chord, you adjust the tone slightly up. If you are on the major third, slightly down. Minor third, slightly up. These rules are consciously applied by choral singers, and are even genre-defining for things like barbershop.

because pianos are real physical things, high and low strings have overtones that do not match the harmonics (and in different ways), so for stuff to "sound good" piano tuners tune on overtones and not fundamentals.

on a more fundamental level our perception of pitch (the musical psychoacoustic thing) also vary with loudness and frequency (physical reality)...

add to that, that at low frequencies we have difficulties tracking the pitch in the first place (so we rely on overtones/harmonics) -- which makes it even more important to tune on evertone (because our heads can recreate/track the note through psychoacoustic effects... so we will ear it "nice" when without the overtones we would be lost)

funny extra fact, our ears are cavities with resonances and distortions... (some musicians exploit that https://en.wikipedia.org/wiki/Maryanne_Amacher )

i might be wrong on some points since i starting learning about music 2 years ago as a hobby but i think it is mostly correct. the whole thing is fascinating.

I suspect there's something to this. I find with guitar you can do something similar if you tune by ear, getting the strings to resonate with each other rather than perfectly matching a tuner. To my ears it produces a richer sound.