The forces between pairs of magnetic dipoles are not central, and in general neither are the chemical bonding forces that hold things together.
That central forces imply that angular momentum is conserved is true. The converse, that angular momentum conservation implies central forces, is not true.
What leads to angular momentum conservation is that that spatial rotations are a symmetry of the laws of nature, and total angular momentum is the associated conserved Noether charge. In other words rotational symmetry implies a "conspiracy" between all the different types of forces so that in an isolated system their net torque is zero.
As an example of such a "conspiracy" consider two bar magnets glued to opposite ends of a ruler. If the magnets are not alligned antiparallel, the forces that each magnet exerts on the other are not central and so they provide a non-zero moment about the centre of the ruler. However the magnets also experience torques that try to make them align antiparallel. Via the glue, these torques are transmitted to the ruler and the total rotational moments about the centre (and indeed about any point) add to zero.