I have not seen 300 so I did not see the coin. ( quite frankly the idea that a leader of the Persians - who were notoriously modest- would be strutting around almost nude with a million piercings was so bizarre I didn't bother).

The process of making coins was quite similar for hundreds of years. A top die and a bottom die were engraved , made from steel, or case-hardened iron, or just iron. This required a craftsman drilling and chiselling out the image with a drill, hammer and graver, then polishing them. Once the dies were done, the bottom die was placed on a solid block or anvil and a weighed piece of copper, silver or gold was placed on it, the top die placed on top of that, and the top die walloped with a hammer. Gold, silver and copper are very malleable, and quite good detail can be gotten from this process. At some point, the bottom die was placed in a ring, that aligned it with the top die, but earlier coins seem to have simply depended on the striker and his assistant to line everything up by eye, as the impressions are often off-center- and so not what you'd call consistent- and the quality of the artwork varied as well: here are examples from the Bactrian kingdoms, c. 200 B.C., that show images could be anywhere from quite fine to perfunctory. As it is far easier to drill a hole than chisel a line, making an image from a series of shallow-drilled holes was the easiest, thus the faces seeming to be made from lots of blobs.

Since they were oblong and not perfectly round, there also was the possibility of such coins being clipped or shaved. So, until perfectly round coins with borders became common after the 16th century, coins still had to be weighed to make sure they were what they seemed.

EDIT Here are some examples of Persian coins.