In classical mechanics, it is true that the dynamical variables can be represented in terms of position and momentum. The terminology « dynamical variable » in fact comes from classical Hamiltonian mechanics. For example, parts of a mechanical system in Hamiltonian or Lagrangian mechanics.
Quantum Mechanics borrowed much of this philosophy from the Hamiltonian formulation of classical mechanics (which is related to the Lagrangian formulation of dynamics, and even Quantum Field Theory borrowed much from Lagrangian Dynamics). But in Quantum Mechanics things have to be more abstract since we have many non-classical quantities such as spin.
In Classical Mechanics, the state of the system is given by the three position coordinates and the three momenta along the coordinate axes of each particle. If there are $n$ particles, this is 6$n$ coordinates for the entire state. These are themselves variables, but any function of these coordinates is also a « dynamic variable » of the system.
In Quantum Mechanics one has a kind of analogy and there is a formal definition in this case: if the space of all states of the quantum system is $H$, a Hilbert space, the dynamic variables are the self-adjoint (Hermitian) operators on $H$. These are no longer functions of the states, but operators on the states. But a function can be thought of as a non-commutative kind of function, or, rather, the non-commutative generalisation of a function, so the passage from classical mechanics to quantum mechanics has often been expressed as the passage from commutative dynamical variables to non-commutative dynamical variables.
The physical meaning, either way, of a dynamical variable is that it is any physical quantity of the state which can be measured. A synonym for « dynamical variable » is « observable ».