There are such things, but they are, as you say, highly unconventional. The only application that I have come across lies within quantum mechanics. The idea is that fundamental particles such as electrons don't have an identity, since they cannot be distinguished even in principle, and so they cannot be uniquely labelled.
There is some discussion of this in the SEP article on Identity and Individuality in Quantum Theory, particularly in section 5, Non-individuality and self-identity. One approach to expressing this in formal logic has been developed by da Costa and Krause in what they call Schrödinger logic, which has restrictions on self-identity.
This example might help to explain it. Let's start by way of contrast with a classical situation. Let's say we have a classical box with a hole in it. We put a marble into the box through the hole. Then we put in another marble, qualitatively indistinguishable from the first. We give the box a good shake and after a while one of the marbles pops out. Then the other pops out. Was the first one out the same as the first one in, or was the first one out the second one in? We don't know because we didn't look inside and track their movements. But each marble has a distinct identity, so there is some definite fact of the matter, we just don't know it.
Now change the example to a quantum box. We put two electrons in, give them the equivalent of a shake, and later they come out. Which one is which? Here, there appears to be no fact of the matter. All we can say is two electrons went in and two came back out. The electrons don't have a unique identity. If we had looked inside the box and measured their movements, that would have been a different experiment.
- da Costa, Newton; Krause, Décio (1994), "Schrödinger logics", Studia Logica, 53 (4): 533–550.
- Krause, Décio; da Costa, Newton (1997), "An Intensional Schrödinger Logic", Notre Dame Journal of Formal Logic, 38 (2): 179–194.