You might use those different pad shapes for marking of some pins, for instance pin 1 of an IC square, all others round. For polarized capacitors you may use different pad shapes for the + and - pin, same for diodes. If you want to use those different shapes for marking use only those easily recognized without a magnifying glass.

When copper pours need to fit through via groups, square vias make it easy to ensure that the copper will join through the gaps based on your design rule set. While actagonla and square via can not do.

The shape of the pad has no functional significance for a plated-through via. Indeed, on high-density boards, the pad around the via is sometimes as close to zero as they can reliably manage within the tolerance of the manufacturing process. So there is no functional difference between different shape of via pads.