Differential equation degree doubt

The explanation is simple: they are not the same equations. Even if two equations are equivalent, they are not exactly the same. For example:

$$\frac{dy}{dx}=\sqrt[3]{x}\tag1$$ $$\left(\frac{dy}{dx}\right)^3=x\tag2$$

The equation $(1)$ is not the same as equation $(2)$ even if they do have exactly the same solutions (in $\mathbb R$ to be clear). You can see that $(1)$ has degree $1$ and $(2)$ has degree $3$.

The problem is when you try to find degree of i.e. $$y=e^{y'} \quad\text{or}\quad y=\sin\left(\frac{dy}{dx}\right)\tag{a,b}$$ There exist a formula that allow you define a degree of non-polynomials, namely $$\deg\;f(x)=\lim_{x\to\infty}\frac{\log|f(x)|}{\log(x)}$$ but in some cases, such as $(b)$, is unlikely to work, whereas for other cases it allows to define a degree of non-polynomial functions. For example equation $(a)$ may be degree $\infty$.