This is actually a very deep question, good job!
If you think about what counting means, it is a sort of song and dance game, where the song is usually a sort of chant that we have just agreed upon, where the important thing is that each sound that you chant is different from every other sound in the sequence. Then the game has certain rules, where in time with the chant you are pointing at objects, and you must never point at the same object twice, and you must point at all of the objects, and then whatever the last word you chanted was, we say is the count of the thing.
You are right to intuit that this game cannot be played in this form with electrons: at least, not according to quantum mechanics. You can point at distinct states that the electrons are occupying, and you can count those, but you cannot count the actual electrons, because they are indistinguishable particles, and there is generally a non-zero probability that they switched places, and this probability actually has some measurable consequences, in that the electrons behave statistically according to Fermi-Dirac statistics.
With that said, the fact that there is no observable difference between any electrons, actually kind of helps us out a little bit. It means that we can speak very loosely about a state occupied by an electron, as “an electron,” and not worry about tracking the electron in that state to see if it changes places with another electron. The only reason we'd really care about them changing places, is if it made some sort of difference to some sort of measurement, and it cannot.
So when we count electrons, or try to label them, we are really labeling the state that they are in, and counting the states that they are in, and confirming that those states are occupied by an electron. But at a deep level indistinguishability does mean that we can't tell you that “it's for sure the same electron occupying this state at time $T+1$ as had been occupying it at time $T$.
Of course, the states occupied by electrons, do have real physical consequences. For example, you can measure the electric charge in a box, and count electrons that way. Or for another quick example, water contains these atoms of hydrogen and oxygen, which if the electrons were separated and not configured so that the electrons were in these shared states between the hydrogen and oxygen atoms, would be flammable. But they're not once they are water.