Is there a fundamental distinction between objects and its types?

"Martin-Lof type theory" is, unfortunately, not a completely unambiguous designator of a type theory. However, in the type theory used in the HoTT book, it is true that types are particular terms, namely terms whose type is a universe. (Of course, not every term is a type.) In other closely related type theories, types are not particular terms; instead there is an operation "El" (for "elements of") which maps terms-whose-type-is-a-universe to types. The first version is sometimes referred to as a "universe a la Russell", and the second (with "El") as a "universe a la Tarski". Russell universes are often more convenient to use, but Tarski universes have some formal advantages, due to the fact that they don't "mix levels" between types and terms.