A stochastic differential equation is nothing more than a short-hand notation for a corresponding integral equation. So the initial SDE you provided actually means
$$ \int_0^t d S_u = \int_0^t \mu(S_u, u) du + \int_0^t\sigma(S_u, u) dW_u$$
This is how the SDE is defined (see e.g. here). The reason is that you cannot differentiate a Brownian motion. It does not have a derivative according to the usual definition of calculus (taking limits etc).
Things like $\frac{\partial y}{\partial S_t}$ just don't make sense in the world of stochastic calculus.
OK, so, back to your equation. Note that it can be written as:
$$ \int_0^t dy_u = \int_0^t S_u du + \int_0^t S_u dW_u$$
with $y_0 = 0$. Then the corresponding SDE is simply obtained by removing the integration signs:
$$dy_t = S_t dt + S_t dW_t$$
That's it! And why? Well, again, because this SDE is actually defined as the corresponding integral equation. There is no corresponding differential equation which involves actual derivatives.