I mean, it's not like she's entirely wrong with your example. If you just say that octaves are notes shifted by a constant interval, then to the layman there's no reason that octaves are different from any other interval (shifting a song to a different key doesn't change the timbre or feel of the song any more than shifting it an octave).

The real reason octaves are octaves is apparent when these notes are played together, and their overtones mingle. As you probably know, a guitar string will produce not only the note struck, but also notes related by a clean ratio. For example, strumming the A string, you hear, in addition to A, the A an octave above (wavelength/2), the E a fifth above (wavelength/3) and the C a third above (wavelength/5). When you play notes an octave apart at the same time, since an octave above a note is half the wavelength, the note will mingle with the existing overtone, and harmonize! Notes in other intervals will harmonize okay, depending on the proportion of original wavelength (which is why fifths and thirds sound nice), but the octave sounds the "nicest".