The reason is that set theory is a first-order theory, and first-order logic is untyped. Namely, there is a universe and everything in the universe is "the same".
It is true, you can introduce predicates in the form of relations, or define such relations given your language (and theory). But the general idea is that the elements of the universe are just that.
In set theory, there are no types and we call all the objects of the universe "sets".
This is one of the reasons some people feel like type theory could provide a better foundations for mathematics, as it reflects more closely some of the naive ways we think about mathematics.
(While it is indeed the case, set theory can internalize type theory in fairly straightforward ways for the most part, so there is no "right" or "wrong" approach, only a matter of convenience. And though set theory is untyped, this is not necessarily a bad thing. Your computer is untyped as well, remember that all the things that you might write in any programming language end up being just a bunch of "pass current" and "block current".)