Why do people say "an octave louder"?

Octave means you have reached the 8th degree of a scale which is identified as a repetition of the tonic.

There is no tonic in a chromatic scale.

The total chromatic (or aggregate[3]) is the set of all twelve pitch classes.

Yes. I know that.

Basically, the point is just a nitpick. A diatonic scale has seven pitch classes . A chromatic has twelve. If you read the original post being responded to, engineer_comp_geek says the Western scale has 12, not 8, tones between octaves if you include the chromatics. There are 5 extra non-diatonic notes between octaves if you include the chromatics. So the statement should either have the numbers 12 and 7 or 13 and 8, depending on whether you want to count the octave. I would go with 12 and 7, and say that Western scales are generally divided in 7 sections, not 8.

For a nitpick you are really digging in.

No tonic means it is meaningless to say there are 13 notes in the scale. There are only 12, and never 13.

In a major scale the 8th note is the cycling of the tonic, thus called an octave.

The eighth note may be called part of the scale because it is the tonic, in other words it has a musical value coming around again, where in the chromatic there is none.

Think about it any way you want.

Either way works.