What is an octave? It is said to be eight, but I cannot see the eight anywhere When I searched for octave, I listened to the fact that if you take an octave band the lower cutoff is double the upper cutoff.

Where is this name octave coming from, that is, where does the figure eight show its significance in the above statement?

In the same way, if the above is clarified, I have some doubts on 1:2 and 1:3 octaves.
This comes from music terminology. The name "octave" comes from the fact that in the heptatonic musical scales (which are the prevalent scales in western music), the note with a 2:1 frequency ratio is the eighth note in the scale.

For example, in the C major scale (C D E F G A B C) the eighth note is one octave above / has a 2:1 frequency ratio with the tonic.