Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

Not exactly. Although $\dfrac{dy}{dx}$dxdy​ looks like a ratio, it is defined as a limit:$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}.$$

So it represents the rate of change of y with respect to x at a point, not a simple ratio of two independent quantities.

In some contexts (like separable differential equations), it can be treated like a ratio, but its meaning comes from this limit definition.