Is Marginal Rate of Substitution a multivariable function?
$\frac{dy}{dx}$ is written as a function of a single variable right?


I don't think so. Your shorthand notation should be standing for

$$\frac{dy}{dx} = -\frac{U_x(x, y)}{U_y(x,y} $$

and hence is a function of both variables ${x,y}$. That makes sense, as your MRS could depend on the locus you're at.