Since 31 is a prime number, selecting either 7 or 12 pitches from 31 to give a diatonic or chromatic scale will inevitably produce an unequal temperament.
Selecting the appropriate pitches from 31TET gives a very good approximation to quarter-comma meantone. The major thirds are only 0.8 cents different. In a pipe organ, there is often acoustic coupling between notes that are playing simultaneously, and that can be enough to "bend" the pitches and give a just intonation (equal to a 1/4 comma meantone) major third. There is no coupling in an electronic simulation, of course, unless you explicitly program it into the simulator.
The difference between minor thirds in the two temperaments is about 0.6 cents, though both these thirds are 5 or 6 cents different from a just intonation minor third.
An intuitive explanation of why this "works out right" is because 31TET divides a (diatonic) whole tone into five parts, not two as in 12TET or four if you use quarter-tones. Splitting the five parts into 2+3 gives a pretty good approximation to the two different sizes of semitones in meantone temperament.
See https://en.wikipedia.org/wiki/31_equal_temperament, and the references on that web page.