Can different variables refer to the same object without an identity rule stated explicitly?

Question

For example, $\forall x(Qx\rightarrow \exists y(Py\wedge Rxy))$, if the Universe of discourse only contained one object, can this sentence be true?

Answer 1

Yes, different variables can refer to the same referent even if equality is not explicitly stated in the sentence in question. For example, in a structure with a single element the sentence $$\forall x\forall y(Px\leftrightarrow Py)$$ is always true.

Answer 2

For example, $\forall x(Qx\rightarrow \exists y(Py\wedge Rxy))$, if the Universe of discourse only contained one object, can this sentence be true?

Certainly. In the universe of exactly one red apple: $\forall x~(x\text{ is an apple}\to\exists y~(y\text{ is red}\wedge x=y))$