In many cultures I have seen in the world circular currency was what they used in medieval times. I have seen very few cultures that used rectangular currency even for a short period of time. The question here: is there a reason that so many cultures used primarily circular currency? And is there a reason why rectangular currency wasn't more popular? Keep in mind I am talking about the medieval times.
To put it plainly, round coins just seem easier to make and work with. Because casting is a pretty complicated and energy-intensive process, coins would usually be stamped instead.
I have here a period illustration of mediaeval minting from the church of St. Barbara in Kutná Hora. What you see on the bottom two pictures is the forming of the blank and stamping of the coin, respectively;
The blanks were cut out of a hammered rectangular sheet of silver by the expedient of holding the sheet over the edge of an anvil and striking it with a hammer (bottom left). This would initially produce smaller rectangles; their corners would then be cut off in the same fashion to produce a round blank.
On the right, the mintmaster is stamping the coin. The (steel) stamp has two parts, both negative; one attached to the anvil below, and the other on a hammer-like object held at the top. The top would be struck with a hammer - you had to do exactly one powerful strike to get a clear imprint.
The reason why round currency was preferred likely have to do with the stamping; it is easier to line up round stamps than rectangular ones (if you misalign them, the images on the rub and the obverse will be slightly rotated relative to each other, but edges are still in the right place), as it is also easier to make them (nice and crisp rectangles are pretty finicky to forge). Additionally, there is something to be said for the coins not having sharp corners.
Individually (or even put together), none of these factors disqualify rectangular coins from existing - indeed, we have examples both from the Orient and from Europe (namely Transylavnia) - but at the end of the day, round coins are just so much more convenient.
Adding to the already good answers, I'd like to toss in a little bit of physics.
As mentioned, coins during the Medieval period were generally hammered/stamped (casting wasn't common because it's harder to do accurately with dense materials such as metal). When hammering something, force is distributed radially out from the center of impact, which lends itself naturally to a round shape. Think of a wad of dough being smashed under a board: The dough will ultimately come out with rounded edges, even if it starts out roughly square to begin with. Corners are simply harder to manufacture through this process, they're much easier when cutting, as in paper money, which didn't exist in Europe at the time.
Note that paper money did exist in parts of Asia as early as ~700CE, and in those cases it was square, probably because cutting straight lines is easier than cutting curved ones.
I think the way money was done is the answer. I really can't translate all the specific words in English, but check Wikipedia. A smooth rectangular coin is more difficult to make.
Medieval money was not very regular with round hammers smashing a piece of metal (I hope that's the good word) but the strength applied on metal by a rectangular hammer would have been irregular and the coins would not have had straight sides. They would have look like a melted pat of butter, unless a big work of carving the sides is done after hammering coins.
The weight of a coin was the best way to prove it was not false, so carving the edges to make them smooth and straight would have lightened the final coin.
- in the middle ages, money was commodity money i.e. gold/silver/bronze coin whose value was the actual value of the coin. Over the 16th-19th centuries, the world moved to paper money (still generally backed by hard assets e.g. gold and silver reserves), and then fiat money
- the main considerations for making commodity-money coins from precious metals (or an alloy of) are a process that doesn't waste metal, cannot be counterfeited (e.g. coating or alloying with lead). Round coins are easy to stamp and and round edges easy to mill/finish; we can melt the wastage. No advantage to a rectangle or polygon, and corners would snag or damage a purse.
- the main considerations for making paper money are something light, strong, tough, convenient to carry, cannot easily be defaced or altered, and again cannot be counterfeited (in modern technologies, i.e. inks, color security patterns, foil strips, translucent watermarks). Something square/rectangular allows it to be printed without waste. Triangular corners would dog-ear. Now why rectangular? Per Why are banknotes rectangular, we want something that can be easily stacked, bundled, counted etc. Rectangular beats square because it self-orients: there are only two possible ways to orient it, and the pictures and text very quickly tell us when we've got it upside-down. You can quickly count the number of bills in a stack by riffling through them. The long-side allows you to print dates, serial numbers, name of issuer etc. Rectangular also beats square in volume-efficiency: they can be folded and kept in a wallet.
- the answer is not due to technological impossibility, as I first expected: in China the Tang Dynasty pioneered block-printed paper banknotes since 740 with inks and anticounterfeiting; until 13th-C devaluation under the Mongol dynasty. The printing press was invented 1410.
- (there was legitimate resistance in switching from commodity money to paper notes in the 17th-18th centuries, and not even primarily due to counterfeiting A History of Printed Money. Citizens would distrust that the note would be redeemable by its issuer for face value, due to inflation, wars etc. Some governments could and would literally print more banknotes, which meant your notes could always be devalued and were only as strong as the reputation for fiscal restraint of the govt that issued them. That addresses why (rectangular) paper money didn't become popular.)
- so why was China 900 years ahead compared to Europe (first European banknote: Sweden 1601)? My instinct is it was lack of stability: the absence of a lasting strong central govt guaranteeing notes with hard assets through all crises, invasions, wars, asset seizures, of which medieval Europe had lots. I think it's no coincidence that European paper money only became accepted after the end of the European Wars of Religion (1648 Peace of Westphalia) when European countries largely agreed to stop invading each other. Also, the Age of Discovery had started, and it was more profitable and less political blowback (in Europe) for European powers to attack each others' New-World colonies and shipments. It would be good to quantify all this.
I have a different opinion, mostly based on school lessons. I'm not sure if it's totally wrong or right, it's just another point of view.
At the beginning, commodity money was rectangular. Larger denominations of money like grivna could be in the form of a stick.
Early coins were made from precious metals and their value was based on their weight, so their form or shape was not as important.
Since weight was important for determining the value of the currency, one issue was coin debasement. It's easy to cut a piece from a rectangular coin so that nobody would notice.