I can't seem to find a definitive answer to this. ExecShirts.com (however famous they are) call it a Small Spread Collar:
The Small Spread Collar is another unique collar, it has very little
spread with the collar halves coming almost straight down. The edges
of the collar are rounded for a softer look. Their is a small space at
the top of the collar for wearing a neck tie. This shirt should be
worn with a neck tie. This collar is sometimes called a "Soprano
Collar" or "Mafia Collar" or even a "Continental Collar" because it is
worn on the European Continent.
GuidoFashions advertise them as being "guido" collar shirts, but don't go into any more specific details than that.
Neither "guido" or "small spread collar" turns up a large amount of results though.
Searching for a classic narrow seems to show some similar collars, where the narrowness seems to vary from shirt to shirt - but even then that terms up wildly different results.
I think the best result I can find is a club collar. It doesn't turn up a large number of results (or matching images), but is referred to as such in a few articles. For example, Vintage Dancer state:
In the early 20’s, the stuffy detachable collar was still the norm.
They were always white, always detached, and quite tall (up to 3
inches.) The round edge club collar was the most fashionable from the
preceding decade. Fans of Boardwalk Empire will recognize Enoch
‘Nucky’ Thompson who is almost always wearing one. The pointed collar
gained favor after 1923. The collar was still tall and the points
longer than today’s dress shirt. Some soft point collars came in the
button down style.
The pointed collar shirt remained popular for the rest of the 1920’s
although other versions such as the spread collar had their famous
moments too. The popularity of the wide Windsor knot tie required
collars to have wide openings hence the introduction of the spread
collar in both round and point styles.
This is the picture they use to describe this:
Ultimately, in this day and age, I think gangster shirt or Mafioso shirt is probably the best way to describe them and certainly turns up the best results.
So it seems that the shirts were indeed worn back in the early 1920s/1930s, but ultimately the popularity of The Goodfellas, being a classic film of the genre, must inspired many imitations in other films.