I guess this is valid for trained musicians. For me, I can't possibly identify the same note in different instruments, as all these sound very different to me.
The only thing I seem to be able to do is to identify a song based on the very first second or two. Because they are different sounds, not just notes.
Not an undertone, it's fundamental frequency. Harmonics/overtones follow a very specific pattern:
https://en.wikipedia.org/wiki/Harmonic_series_(music)
...so in most cases there isn't really any ambiguity on what the actual fundamental frequency is. The best way to demonstrate it is probably with a high-pass filter (aka. low-cut filter). High-pass filter allows only higher frequencies to pass through, or in other words it cuts out lower frequencies. On the example below you can hear that as the lower frequencies are getting removed from the signal, it's still the same note:
https://www.youtube.com/watch?v=50lRE2Bgag0
I wish the filter sweeps were slower, but that's the best example I could find
It's not even about which frequency is dominant. I'm not an expert on psychoacoustics from the "sciency" side, but basically our brains can figure out the fundamental note based on the harmonics alone, even if we don't hear the fundamental frequency itself. De-emphasizing the fundamental frequency is sometimes used in synthesis just to create an interesting sound, but most notably this phenomenon is heavily relied on in modern heavy bass electronic music and down-tuned metal. A really common trick is to distort the bass track, so it can be still heard even if speakers can't reproduce that frequency, as distortion generates more harmonics. I don't mean distortion in literal sense, but in audio effect sense, so effects like saturation, overdrive, fuzz, clipping, etc.