Suppose I have two small parameters $\epsilon$ and $\delta$ in some perturbation problem: $0< \epsilon <<1$ and $0<\delta<<1$, with $\epsilon$ and $\delta$ the same order of magnitude, $\epsilon = O(\delta)$. Is it correct to state
$$\frac{\epsilon}{\delta} = c$$
Where $c$ is some constant, with $c = O(1)$?
No, it is not correct. It is correct that $\frac{\epsilon}{\delta} \le c$ for some constant $c$.
The assertion "Where $c$ is some constant, with $c = O(1)$?" is redundant. Any constant is automatically $O(1)$.
You could say that $\epsilon/\delta = O(1) $ though.