$\mathbf F$ and $\mathbf B$ are indeed in the same direction. Remember that in the equation you quote the (imaginary) thing on which the force acts is not a moving charge but a (North) monopole.
Think about a compass needle (pivoted magnet). It points with its North end in the direction of the external magnetic field, and its South end in the opposite direction. An early theoretical model of a magnet was a separated pair of equal and opposite monopoles, one near each end. $\mathbf F=m\mathbf B$, with $m$ as pole strength, would account for the equal and opposite forces experienced by the ends of the needle, and the direction in which the needle points when it has stopped oscillating. What's more, if each monopole gives rise to an inverse square law field (one inwards, one outwards), it also accounts quite nicely (using vector addition of fields) for the shape of a magnet's field outside the magnet. But it gives exactly the wrong direction for the field inside the magnet! And it's a bit disconcerting that no-one has yet found a monopole by itself.
The theory has long since been replaced by the idea that a magnet's magnetic properties arise electrically from spinning electrons. A current-carrying solenoid is a better model of a magnet than a pair of monopoles. Nonetheless, as mentioned earlier, there is a tempting partial fit between the monopoles model and what we observe, and at least until the mid twentieth century many courses in magnetism and electromagnetism made use of the idea of monopoles, at least in the early stages. Messy patches were used to compensate for the wrong field direction inside magnets and magnetic materials when treated as collections of true dipoles (as opposed to small current loops).
You ask about the accuracy of the inverse square law for magnetic poles. I believe that Coulomb used a torsion balance with ball-ended magnets, which behave approximately as if their poles were situated at the centres of the spheres. This method would not be capable of great accuracy. Because poles always come in pairs, there's no magnetic analogue of Maxwell's very accurate hollow sphere method for demonstrating the inverse square law for charges. But... If we use magnetic poles as a theoretical construct, we can obtain some results that are as accurate as we like. In particular, that a long way from it, the magnetic field pattern of a current loop is exactly the same as that of a dipole consisting of equal North and South monopoles on the loop axis, separated by a small distance. The Wiki article on 'Magnetic Dipoles' gives more detail.
