Well, in principle, it can happen that two particles would obey this statistics, and be different in some subtle way, and the statistics would be broken if that subtle difference is allowed to interact with the environment, but not before. I think you can see how it can happen under MWI. Statistics is affected not by whenever particles are 'truly identical' but by whenever they would have interacted with you in identical way so far (including interactions with environment - you don't have to actually measure this - hitting the wall works just fine).

Furthermore, two electrons are not identical because they are in different positions and/or have different spins ('are in different states'). One got to very carefully define what 'two electrons' mean. The language is made for discussing real world items, and has a lot of built in assumptions, that do not hold in QM.

edit: QFT is a good way to see it. A particle is a set of coupled excitations in fields. Particle can be coupled interaction of excitations in fields A B C D ... and the other can be A B C D E where the E makes very little difference. E.g. protons and neutrons, are very similar except for the charge. Under interactions that don't distinguish E, the particles behave as if they got statistics as if they were identical.