When one says that an elementary particle is point-like, one is referring to the fact that theoretically, there's no limit to how small a region a detector can localize a particle to. For the sake of argument, let's imagine two wrong things (a) that such an ideal detector is possible and (b) complications arising from Planck scale physics don't change anything conceptually.
Even if you allow for that, your worry that information is being stored in a zero volume region is still unfounded. It would be a legitimate worry in pre-relativistic-QFT physics. But we know particles aren't pellets that move around carrying information. They can disappear and be spontaneously created out of the vacuum. What this is hinting at (though some might prefer a different picture) is that particles aren't fundamental - fields are.
The quantum fields for various particles are defined everywhere in space. Once you specify what kind of structure the quantum field is (a scalar, vector, spinor, etc.) and what its other properties are (say, the symmetry group under which it has local gauge invariance), you have specified what spin, charge, mass etc. its particle excitations will carry. Since the field is defined everywhere in space, there's plenty of room for all that information. So in a certain sense, the information that a detector detects is encoded everywhere in space (because the field is everywhere) - and the specific structure of the detector just picks out the right information you asked for.
Finally, two point-particles (say an electron and a muon) have different properties because they are excitations of two different fields defined everywhere in space - and the detector you build specifically for the electron will pick out the "signal" from the electron.