I read chapter 6.12/6.13 in J. D. Jackson's Classical Electrodynamics about the magnetic monopole and a certain detail is confusing me.

First in a semiclassical consideration of a magnetic monopole, the fact that change of angular momentum must occur in integral multiples of $\hbar$ is used to show that magnetic and electric charge must have discrete values.

Then a simplified discussion is given of Dirac's original argument that leads to the same quantisation condition for electric and magnetic charge. But in this presentation - when I understand it correctly - the quantisation of angular momentum is not used. Instead, single-valuedness of wave functions is used, together with gauge invariance.

So I am not sure, whether quantisation of angular momentum is really needed to find the quantisation condition?

Of course when not, the quantisation condition for electric/magnetic charge would explain the quantisation of angular momentum, wouldn't it?