From a physics perspective...
Different pickups have different magnetic fields. The magnetic fields have different shapes and sizes.
Part of what makes typical humbuckers sound different from single-coil pickups is the fact that the magnetic field on a humbucker covers more of the string. When it covers more of the string, that means that the string's higher modes (higher frequencies) will cancel out more. This is part of why typical single-coil pickups sound brighter than P90s, and P90s typically sound brighter than humbuckers.
Another factor that makes pickups sound different is the overall frequency response of the pickup. When you add more windings to a pickup, the output levels get higher, but it also increases the inductance and parasitic capacitance of the pickup, which makes the pickup sound darker. The wire also has resistance to it, and a higher resistance (thinner wire) will dampen the natural resonance of the pickup (reduce the Q factor).
In theory, you can adjust for the frequency response with EQ, but you cannot use EQ to adjust how big the magnetic field is. The problem is that making the magnetic field bigger, like for a humbucker, has a different effect on each string.
The reason why it affects each string differently is because you can think of it kind of like an EQ, except instead of applying an EQ to different frequencies, the EQ applies to different wavelengths. For example, if you have a magnetic field that is 2cm wide, then it will pick up a wavelength of 8cm just fine, but a wavelength of 2cm will be highly attenuated.
Each string has a different relationship between wavelength and frequency. So, changing the shape of the magnetic field is kind of like applying a different EQ curve to each string! Once all the strings are mixed together, you can't do that. That is why it is not possible to make a humbucker sound like a single coil pickup, or vice versa, using EQ.
Now consider that different pickups have lots of different magnetic fields. For example, you can get pickups with pole magnets or bar magnets. Also consider that when a string vibrates, it moves through different parts of the magnetic field... if you hit a string hard, it won't line up with the pole pieces, but with a bar magnet, that doesn't apply. These are differences that you can't just EQ out.