A → BC, or A → a, or S → ε,
Above is the typical definition given on Wikipedia and in textbooks. Nothing i can find says the B and C variables have to be different variables either, but nothing i can find says they can be same definitively . I am leaning towards it being valid but i just want to make sure.
They do not have to be different variables. See, for instance, the final example in this PDF, in which a grammar is reduced to a Chomsky normal form that includes a production $X_{aa}\to X_aX_a$.