Can the magnetic force between two magnets be explained classically via magnetization current?

Usually in Electrodynamics courses the magnetic forces are analysed via wire-wire interaction. I don't remember being shown a classical explanation for magnetic forces between magnets. Since it is a quantum system caused mainly by the alignment of spins/magnetic moments, I was wondering if the magnetic force between two magnetized ferromagnets can be explained via Lorentz force on the small magnetization currents formed within the material (even if they are just a mathematical tool in this case).

Yes. The quantum-mechanical origin of the magnetization is largely irrelevant: all you need to know is the material's magnetization field $\mathbf M(\mathbf r)$, which is defined to be the (locally averaged) total magnetic dipole moment per unit volume at and near position $\mathbf r$. The magnetization then determines the magnet's magnetic field outside it and therefore its interactions with its neighbours. Similarly, the magnetization current $\mathbf J_m=\nabla\times\mathbf M$ (coupled with the magnetization surface current $\mathbf K_m=\mathbf M\times\hat{\mathbf n}$) is a perfectly adequate description of the currents that cause the field of the magnet as well as its interaction with the fields of other magnets.
This assumes, of course, that the magnetization itself is fixed. In reality, $\mathbf M$ is a complicated quantity, and it depends on the applied magnetic field, the temperature of the material, and the history of both those quantities. For paramagnetic and diamagnetic materials, the linear relations $\mathbf M\propto\mathbf B$ mean that the magnetization currents in a material will change depending on the applied electric field. For a ferromagnetic material, you need to refer to the quantum mechanical description to find out whether they will do so and to what extent.
As a good zeroth-order approximation, though, you can assume that the magnetization is frozen in the material, with the magnetization currents perpetually going around the magnet, and calculate the forces classically based on that. If you want more detail, you can ask the quantum theory for the hysteresis curves of the magnet and then use those to determine what the changes will be. The framework, however, remains that of classical electromagnetism together with a definite constitutive relation between $\mathbf M$ and $\mathbf B$ (and potentially $T$ and the material's history), which is provided by quantum theory but used on its own after it has been specified.