It is, I think, a perfectly clear observation that one note an octave
above another note sounds as if it were the same in a certain sense.
It's certainly common for people to perceive things that way, but it's not universal. Here's a question from someone who complains that they don't hear things that way, for example!
shared harmonics alone can't be seen as a definitive reason to see an octave as special...
- ...for the reason that you pointed out (that for a timbre with all overtones, a higher note with a fundamental frequency that is the same as that of one of the lower note's higher (>2) harmonics will also have a subset of the lower note's harmonics)
- ...because for sounds that don't have the full set of overtones, the 'subset' thing doesn't work - a sound with only the first, third, and fifth harmonics won't share any component pitches with the same timbre sounded an octave up. This is the case for a closed-pipe instrument, like a clarinet, though it's worth pointing out that most instruments do have both odd and even harmonics present.
The idea also doesn't work for octaves for sounds with enharmonic partials, although cultures that use such sounds (e.g. Javanese) often use different scales - so this could be seen as an exception that proves the rule.
while they are by no means the same exact note, they are named with
the same letter
We have to remember that the letters are a culturally-specific thing. The reason that notes an octave apart have the same letter is closely related to the fact that Western music culture assumes an octave-repeating scale. You don't have to have an octave repeating scale...
Octave equivalency is a part of most "advanced musical cultures", but
is far from universal in "primitive" and early music. The
languages in which the oldest extant written documents on tuning are
written, Sumerian and Akkadian, have no known word for "octave".
...but the strength of the octave relationship means that an octave repeating scale tends to work well for more harmonically sophisticated music where groups of notes have to sound good together. For example, if we consider a base note C3, the fifth (G4) up from the octave (C4) also itself has an octave relationship with G3, which in turn has a strong relationship with C3. If you had a scale that repeated around a 3:1 ratio rather than 2:1, I don't think things would be as 'tight'.
Also, while your 'octave plus a fifth' relationship clearly doesn't have octave equivalence, the next harmonic would be a two-octave relationship - again, this isn't enough to say that the octave is qualitatively and distinctly special, but it does point to the strength of the octave compared to other ratios.
A natural occurrence of the octave as somewhat 'special' is with a flute-like instrument, where blowing harder gets you up an octave higher with each fingering - maybe this could be an influence on the adoption of octave repeating scales too.
one can for example take any piece of music consisting of two parts
and translate one part an octave up, leaving the other the same, and
the piece will still work.
Probably subjectively true in many cases, but as Kilian Foth pointed out in the comment, there are cases where it may not subjectively work as well, depending on the voicings, harmonic movements, and timbres involved.
In summary, I don't think you can say that an octave relationship displays an objective qualitative 'equivalence' that another simple ratio doesn't. It's more the fact that the octave relationship is stronger than other relationships that leads us to the idea of the octave-repeating scale, and to commonly-perceived subjective equivalence of the octave.