The answer to this question is yes, in music, octave is indeed a synonym for distance. It refers to a specific scale interval between two notes, which is the same when both notes are raised or lowered by one or more octaves.
An octave is often used to describe the range of notes in a particular scale. For instance, a scale may be referred to as a C Major Octave, which would include all of the notes from C to the next C an octave higher.
In some cultures, an octave also refers to the distance between two notes that are the same but with different tonal qualities. For instance, between a low and high C, the difference in tonal quality is one octave.
In music theory, octave is used to describe pitch intervals. It is defined as either a higher or lower note that has either twice or half the frequency of a given note. It is also used to describe the one-octave (or two-octave) transposition of a melody or chord progression.
Octave can also refer to the direction in which tones move in a melody. An ascending octave would refer to tones that get higher in pitch, while a descending octave would refer to tones that get lower in pitch.
So, to answer the question posed in the title, yes, octave is a synonym for distance. It is used to describe the range of notes in a particular scale, the difference in tonal quality between two notes, and the direction in which tones move in a melody.