The thing you should know is that in QFT, the concept of a particle is a bit different. A particle is a localized excitation of a quantum field. A quantum field

The thing you should know is that in QFT, the concept of a particle is a bit different. A particle is a localized excitation of a quantum field. A quantum field is like a bucket of water, and a a particle is a bit like a propagating wave that you can create that travels "on the field" but it's "made of the field". A mass of a particle tells you how hard you need to hit the field to create this elementary excitation. And this is where the analogy breaks, since a classical field doesn't have this "quantization" of energy, you can create arbitrarily small waves. Whereas in a quantum field, there's a minimum energy that you need to put to excite it and make an electron show up, and that is the mass. It also acts a bit like the (inverse) width of the wave - the higher the mass the smaller the width and thus the more spacially localized it is. Technically the statement that all electrons are the same is the "correct" one, because technically there's only one "state of the electron field" which is described by a bunch of excitations here and there. So the total mass (or more precisely, the energy state that the field occupies) is actually a huge mess that happens to be easily factorizeable into a bunch of localized states. This is only possible because interactions are extremely weak and thus we can just "sum the electrons". But if all of a sudden the electron charge were to become massive, to the point where electrons would start interacting at much bigger distances, the concept of individual electrons would start to become complicated and we'd need to think about them in some other way (like what happens with quarks and gluons)