y common occurrence with acoustic stringed instruments and something we test for when purchasing an expensive instrument. You check for buzzing, dead spots, and RESONANCE. You want resonance to some degree as that adds to the sound but you don't want anomalous resonance which might show up as Bb4 is always 3dB louder than any other note (just a silly example but not impossible).
This brings me to an important point. That the rest of the instrument will vibrate in sympathetic resonance to the note being played and its harmonics.
The harmonic content in the string depends on the attack. Not all strings are the same. In fact one could argue that this is the most important part of sound and the hardest part of learning an instrument, learning proper attack and for guitar learning variety of attacks. Each attack produces a completely different "tone". This makes the guitar a great mimic and has a reputation for versatility. In contrast your piano hammers are fixed. You can control amplitude (strength of attack) and with pedals you can control sustain but you cannot control the initial attack profile of the string(s). Keep in mind that each "key" strikes several strings not just one.
Now typically (but not always) the fundamental is the strongest note, has the highest amplitude or volume among the spectrum of the string. And linear systems do NOT excite sub-harmonics. They don't even excite harmonics for that matter. The other strings will vibrate in sympathetic resonance to the string you play but only if the harmonics of the string are present in the one you played. And they will only vibrate at the frequency of that harmonic. A caveat to this is that coupling with other parts of the instrument could cause coupling between different modes due to a non-linearity, perhaps a joint in the wood etc. thus causing coupling between harmonics. But for the most part the linear model works well. For example if I play the high E string on my guitar, and assuming I attack it so that only the fundamental is present (close to possible if you use your thumb at the 12th fret) then that E will cause the following resonances in the other strings, n = 4 on the low E string, n = 3 of the A string, nothing noticeable on the other strings even though E might be close to a harmonic for some. The presence of these extra notes will add to the volume of the note plucked. As for sustain, you might think that since these are all the same frequency that they would all suffer the same damping. This is true. But you are judging the "decay" of the note by whether you hear it or not and the added amplitude means that the sound will not drop below detection threshold for a longer time. In contrast if the low E string is excited the same way it will NOT cause sympathetic resonance in the other strings. It will be less audible than its higher pitch counterpart.
This brings us to another point. If you are using your ear to make this judgement I don't trust any of it. The human ear is highly non linear in both amplitude and frequency. Our ears create harmonics from the input. This means that even if the higher harmonics are NOT present in sound YOUR EAR WILL HEAR THEM. There is no way that the physics of the instrument can change this. The ear+brain system hears higher frequencies better than lower frequencies to some extent, possibly related to the last point. Bass and treble notes played at the same driving force will be judged as having different volume by listeners. For a Bass note at 100Hz and a high note at 2000Hz both played pianissimo the Bass note may not be heard by anyone. So any claim made about hearing low pitch notes for a longer time is suspicious without more information.
I can say that on the guitar it is simply not true that the higher pitch notes die quicker than the lower pitch notes. Of course there are too many variables to make any answer to this question complete and absolute. If you are really interested in the behavior of the musical instrument and your own ear for that matter, each variable needs to be isolated and the cause and effect 