We can find quite a few reasons for the roundness of coins–these relate to different time periods, but interestingly there are actually several examples of square-shaped coins. However, the round coin is definitely more common no matter what country and history we’re talking about.
The most obvious reason for coins being round is their ease of use, so to say. It is easy to pick up round coins and scoop them up, put them into tightly packed bags, etc, without any worry of corners causing issues. We can also stack up the coins and wrap them up in rolls without worrying about any corners tearing at the package.
Then there’s the fact that coins are intended to be hard-wearing and semi-permanent. Currently, there are several coins still in circulation that are likely older than you or me. This is mostly because of their round shape, which is less likely to wear down unevenly. With a circular shape, no one part of it sticks out further than the rest from any angle, so there’s little chance of any part wearing down. The numbers and imprints might be smoothed out over time, but it would require some very hard usage to erase them completely.
A historical reason for the roundness is that a lot of coins were made by pressing down a small ingot with a pre-made stamp. We can observe a similar process in the old-fashioned method of wax seals. This process typically forces the metal into a rounded shape anyway and means less work to finish the coin. As coins had to exist in relatively large quantities, keeping them a round shape was best for all concerned.
When round coins are cut out, there is some metal left over every time. In a modern-day setting, one might argue that cutting coins into squares from a metal sheet would be more efficient. However, it’s not too much extra work to take the leftover metal and melt it down to use in the next batch.
Finally, a key modern reason for rounded coins is that round coins are convenient for transport within vending machines, etc, since they will roll easily. The same goes for arcade video games and any other machine that requires payment through coins.
No matter what their shape might be, coins have played an important role in the history of many countries.