Just wondering if there exist any equivalence relations that don't actually involve some sort of equality between the pairs of elements in the equivalence relation set.
Equivalence relations involving some sort of equality are easy to construct. For example, the relation over the reals that relates all numbers that are some rational multiple of another number is an equivalence relation (reflexive by multiplication by 1, symmetric by reciprocal multiples, and transitive by multiples multiplied by multiples). Basically, if some fractional amount of some number is EQUAL to some other number, those two numbers can be regarded as equivalent in this perspective (interesting graph theory oriented observations here too).
Can we define some relation from and to some set of objects where two objects are "equivalent" with actually no characteristic in common? If there does not exist any such equivalence relation, then these relations were aptly named, unlike the "closed" and "open" properties of sets. And also, if such a relation doesn't exist, is there at least some equivalence relation that specifies equivalence between objects that are not intuitively equivalent?