A usual dynamical system can be described by an ordinary differential equation. Control systems have a variable $u$ which is to be chosen to get the desired dynamic behaviour.
Once this $u$ is plugged in, whether it is an open- or closed-loop system, do we end up with a common dynamical system just as in the theory of differential equations?
If, with "usual/common dynamical system", you mean a set of ordinary differential equations (ODEs), then in general no. Because it depends on the control $u$. If you have an uncontrolled dynamical system ($u=0$) represented by an ODE
$$ \dot{x} = f(x, u) $$
then it depends on your choice of $u$, if the controlled system is still an ODE.
For example, you could choose a $u$ that contains time delays. Then, the closed loop system would be a functional differential equation (FDE), not an ODE anymore. Or you could choose a $u$ that contains some randomness, then you would get a stochastic differential equation (SDE), and so on.
So you have to choose $u$ "right" in order to get an ODE again.