As previously mentioned, a pickup can't be dirty as such, it can just be coloured in sound. EMGs are very clean and hifi because they aren't wound that much and the magnets aren't strong, so they don't intefere with the string much or the signal. As a result they are very weak, but that is ok as they have preamps in the pickup to boost that weak signal. So they sound strong and hifi.
At the other end of the scale, you get like the dimarzio super distortions, were designed to have such a high output that they distort your gain circuit and thus come up wtih a dirty sound - its the amp that makes the dirt there.
As a result of a lot of winding and a strong magnetic field the eq is very coloured, so they sound quite distinctive
Pickups do sound inherently different to each other. The eq curve, level and impedance of a pickup depends on its magnetic strength, the size and amount of wiring and the style of wiring, so plugged into a non ideal amplifier, as they all all, they will sound different. Obviously any EQ that a pickup has, you could correct for afterwards so it is possible to make them sound similar (for a single coil) although for multicoils on the same string (humbucker or two pickups at the same time) probably never the same.