I'd love to know how they do that. I find it very hard to believe one (or a few) companies have discovered a way to make characterless pickups.
Increasing magnet strength increases the sensitivity, but potentially interacts with the string/strings. So older, weaker magnet types tend to be more neutral.
Coils have two key properties - inductance, related to the number of coils, and resistance, related to the number of coils and wire gauge.
Coil placement - obviously.
Coil physical size also has an effect, wider coils cover more string, changing the harmonic content as they 'average' the signal over a longer length of string.
Varying these factors will change the 'tonal character' of the sound.
I suppose you could use individual pickups per string, small, very low number of windings to minimise the contribution of the pickup. But these would still only tell you what one point on the string was doing, and would have a poor signal-to-noise ratio because of the additional gain required.