"Tradition" is probably the best short answer.
Way back when coinage was invented by the ancient Greeks, they typically used profile portraiture. You'd have to find an ancient Greek and ask them why, but I suspect it was the cultural concept that even functional objects like coins should be beautiful, and they thought the human face looked more beautiful from the side than front-facing. A few ancient Greek states experimented with three-quarter-views, such as the coin from Rhodes I use for my avatar, but as a rule they seemed to reserve full-front-view portraits for figures that were intentionally supposed to look ugly - such as the
gorgon-head on the coins of numerous cities.
The Romans were next on the scene, and copied the concept of side-view portraits off of their Greek neighbours. Once the Roman Empire collapsed, any monarch or republic in Europe that wanted to imply that they were the New Roman Empire copied the Roman style of coinage, including Latin legends and profile portraits. Thus, we have profile portraits on coins down to the present day.
For early paper money, the portraits are usually copied off canvas paintings or similar artwork, which followed the fashion for such artwork of the period, which typically seems to have been half to three-quarters facing.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis