What does my calculus textbook imply that differentials can't be manipulated algebraically?

The textbook defines differentials like this.

Let $y=f(x)$ be a differentiable function of $x$. The differential of $x$ (denoted by $dx$) is any nonzero real number. The differential of $y$ (denoted by $dy$) is equal to $f'(x)dx$.

It goes on to say that the derivative rules can be written in differential form using Leibniz notation. For example, it says the chain rule in differential form is

$$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$

The book says it appears to be true because the $du$'s would divide out, and although the reasoning is incorrect, it helps you remember the chain rule.

Why is the reasoning incorrect? Given those definitions of differentials, what's stopping you from manipulating them algebraically?