> I would love the field of mathematics if it was not so ambiguous.

IME the ambiguity is usually intentional. Take dy/dx for instance. In some sense, d/dx is acting as an operator (or "function" to use programming terminology) acting on the function y to produce y'. However, the intuition here is that we're calculating a slope (ie. a ratio), and the particular ratio we want is the limit of [a sequence of] Δy/Δx. Both of these viewpoints are important to keep in mind to understand calculus, and the point of the notation is to aid that intuition. There are, as I'm sure you're aware, alternative notations for derivatives, and the d/dx notation survives because it suggests things beyond its formal definition.

I think this becomes the most clear when we consider the chain rule, dx/dy * dy/dz = dx/dz. We can't just "cancel out" the dys (outside of exotic approaches such as nonstandard analysis), but allowing that sort of visual pun in our notation makes the formula easier to remember and gives a sense as to why it might be true. And, if we didn't know the chain rule ahead of time, it would likely prompt us to try and prove it in the first place.

The bigger story here is that a lot of mathematics runs on human intuition, and finding a notation which communicates and facilitates that intuition is quite important.