First, we accept that the even tempered nature of a fretted instrument is a compromise, but if you have the octaves right, then every note in between will sound acceptable.
As @paul_c2 says, if you use a bass/guitar tuner (which uses even temper) you can set the intonation at any fret.
If you do it by ear at a non-octave fret, you may well find some notes sound more out of tune, especially in some keys.
On the mechanics, we need to set the intonation because fretting slightly lengthens the strings and slightly increases the tension. The tension increase dominates making the note sharp. We make the distance from nut to bridge slightly less than twice the distance from twelfth fret to bridge to compensate for this. The effect is more marked with thicker strings, partly because we set them higher and partly because they are, err, thicker and therefore stiffer. So the saddles for thicker strings need to be further away. On a guitar changing from plain to wrapped strings makes them more flexible, which is why you see a staggered line.
Finally, strings are not 100% consistent... so swapping in an 'identical' set of strings amy still require a tweak.