Short version of it is most of these names originated a century or longer ago in various parts of the world using different standards, the names have just been kept the same or shortened to the form we have today.

Take 9mm for example. Properly, it should be known as 9x19mm Parabellum, and was designed at the very start of the 1900s. There's a billion different "9mm" cartridges, some even with 9mm in the name, but 9x19mm Parabellum is so common everyone just calls it 9mm, because any other 9mm cartridge you could get is rare enough you have to special order them (like 9x18mm Makarov).

It was designed in Europe, so it uses mm. It has the same bullet diameter, but because .357 was designed in the US, it uses fraction of an inch in its name. At this point, they're essentially just used as names and not measurements, so they stay how they are. The distinction is important because they're completely different rounds and are not interchangeable, here's a pic of them side by sode:

One is thousands of an inch, which is a typical unit of measure for milling operations. As the other person in this chain has said, ballistics is a weird game. End of the day, m(ass)*v(elocity)=f(orce), and the goal is getting f to a useful value. At that level, big and slow and small and fast have the same outcome. But, because ballistics is bastard, other factors start to interfere.

For example, If you want to shoot farther, you can either go faster, or be lighter. Going faster requires more powder in the case, which creates a different profile for the round and changes the acceleration profile during firing (there are several rifle rounds over the years that have approximately the same load and bullet, but are ever so slightly different shaped, to change how the powder burns off and how it accelerates down the barrel)

Being lighter reduces your final impact force, and can impact stability mid flight (leading to alterations to the length and profile of the round itself)

Closer to the first point and tangentially related, the US has been a capitalist hellscape for a while, and has numerous instances of a company making a firearm using a new round because they did the math and said this is what we want and what you should want, followed by everyone else making more or less a clone of it if the gun in question sells well. Due to copyright it can't be identical, but as math goes, there a lots of ways to reach the same outcome.

While the diameter is the same the heights and sometimes shape of the the bullet is a bit different. A .357 is significantly taller than a 9mm. This prevents you from using the bullets interchangeably even though the diameter is the same.