[nandemo]

Agreed. Saying they're the "same note" is an oversimplification. They're "equivalent" in certain contexts, which is an important distinction.

They aren't the same note: if someone plays A3 (fundamental is 220Hz) and then A4 (440Hz) you can easily tell them apart! In contrast, if they play 220Hz and then 221Hz an untrained listener will most likely hear them as the same note.

In some contexts a note and its octave can be thought of as equivalent: if you are playing a chord you can often replace A3 with A4 (and vice-versa) without changing the chord's feel and function. However, that isn't always true. For example, playing G#3 and A3 at the same time will sound more dissonant than playing G#3 and A4 (a minor 2nd is more dissonant than a minor 9th).