As a partial answer to your question, here's an interesting page
on the history of the Massachusetts plates. The first one issued by the state bore the single digit 1
, so it could be relatively small. Soon two digits were needed, then three, so the plates got wider. By 1908 (only five years after the 1
plate was issued, five-digit numbers were needed, and the tag looked somewhat like the European plates you mention.
By the late 1920's, Massachusetts plates were looking more like the standard American tags of today, but were still issued in varying sizes. Not until 1957 were the dimensions standardized (note the centered 3
in the plate pictured on the linked page). Obviously it's easier to make the "blanks" all with the same length and width than to produce a handful of one-digit plates, about 90 two-digit plates, etc.
Although most of the world's plates are rectangular, some interesting exceptions exist. To publicize the wildlife of the Arctic, Canada's Northwest Territories plates are shaped like bears.
In the 1950's, the USA's Tennessee and Kansas both issued plates in the shape of the respective state maps.