While I don’t think the terminology is explicitly standardized, I think most people in the relevant fields would call that statement of the ideal gas law an equation but not a formula, the latter being a special case of the former.

As a trained theoretical physicist, I am entirely unaware of the distinction you are trying to draw here.

Can you find any reference to the idea gas law as a “formula?” As far as I can tell, equations without a single variable on the left side are referred to as simply equations, while solutions of such equations in terms of one variable are referred to as formulas. This seems to be the case for every well-known identity I can think of, like the quadratic equation/formula. Can you think of any counterexamples?