I see that on the various listening tests that the results for each encoder are on a 1-5 scale. Is this value subjectively assigned by the listener in an Olympic judge type fashion or is it a mathematical result determined by ABXing the file against the lossless original?

Subjectively assigned by the listener.

Using the ABX results doesn't work because the listener has to grade the annoyingness of the distortion. A distortion may be easy to hear but not very much objectionable. Also, if you're testing low bitrates where each codec fails, you'd no longer be able to grade.

The subjectiveness of the listener is filtered out eventually by using a sufficient amount of them and averaging the scores.

To clarify a very basic point, ABX tests are not quality tests, but instead they are same/different tests. The whole idea of ABX tests is to determine whether or not its worth the trouble to even bother with tests related to quality. If you can't hear the difference between two sounds, then all questions of sound quality are completely moot.

ABX testing is only relevant in these days because there is so much in audio that actually makes no audible difference at all, yet poorly-informed people keep prattling on and on and on... about sound quality differences.