Fundamental particles identical or just differences very small? I have read that all electrons are identical and I assume the same for protons, etc. That is, all electrons are of the exact same mass and charge.
But do we really know this? Could there be slightly heavier electrons with slightly higher charge or is there a reason why such particles could not exist or would be unstable?
 A: The identity of fundamental particles is a basic hypothesis of all theories, but we don't really know this, neither can we disprove it. Just like the homogeneity and isotropy of spacetime. These assumptions are good starting points of physics theories and can make things harmonic, simple and elegant.
Related link: https://www.quora.com/Are-fundamental-particles-of-the-same-type-all-identical-to-each-other-If-so-to-what-precision-and-why-should-that-be-the-case
A: A particle is just an excitation of its own quantum field. For example, an electron is just an excitation of the electron field. Same for other fundamental particles.
The properties of these particles are intrinsically depends upon the behaviour and interation of respective quantum field with the other quantum fields. For example, an electron's mass is intrinsically affected by the coupling of electron field with the Higgs field and Higgs VEV. If the VEV and/or coupling of the Higgs field does depends on space-time coordinates (i.e., if it's not 246GeV everywhere and everytime?) and vary throughout the space-time than the mass of the fundamental particles like electron would be different place to place and time to time. See the point is the properties of a particle which depends on interactions of two or more fields also depends on the properties of participating fields. Higgs field has been hypothesized to have a constant 246GeV VEV which in turns says that every excitation of a field will have the identical mass, leaving all electrons with identical mass.
A: We have arrived at the standard model of particle physics which encapsulates all particle data up to now. In the standard model the basic hypothesis is that the particles in the table are point particles, with specific quantum numbers.

Thus by construction of the theory there is no way to distinguish one elementary particle of the same type of particle, as they have no extent in spacetime to be able to do so.
When one starts with composites of elementary  particles, as the proton and neutron, because they are in a specific quantum mechanical bound state, they have to be indistinguishable under the same boundary conditions. 
It is only at the level of complexity of solid state that matter can be distinguished into identifiable characteristics, the ones that lead to DNA identification.
So no, there could not be different mass electrons or other elementary particles as long as the standard model holds. The same holds true for composites of these elementary particles. 
Of course history of physics shows that when new frontiers are reached, unexpected  effects may appear, but at the moment the standard model has not been falsified. 
