Yes it was very common, not so much by the time the Vietnam War came around, but certainly in World Wars. Most European Army units were formed regionally, the British and Germans come to mind the most. Oftentimes these units even had the name of their region in the name of their outfit, like the Gordon Highlanders, Seaforth etc... This was a huge during World War I, because so many young men from the same villages and towns would die together. This is how most militaries were organized in past. Regular infantry units were organized and made up of people from the same towns, counties, shires etc...
The American Army was somewhat similarly constructed, except the regular Army Divisions were made up of mostly volunteers (until the draft came about) and soldiers were pulled from all over the country to fill the divisions. That's why in classic World War II movies focusing on the Americans you always have the troupe of the Southern Bumpkin who is usually really Christian but is a crack shot, you have the smooth talking sly New Yorker of Italian descent, the tough and rugged Platoon Sargent from Texas, and the Commanding Officer from some middle class to upper middle background (usually from the North East or the Mid West). That's a stereotypical cliche for most American centered World War II movies, but that was based a bit in truth.
However, the American Army was also comprised of the National Guard. These were State / Regional military units. Pre - War / Early War these National Guard units would pull people from the same towns and counties to make up companies. So often times you had a bunch of guys who grew up with eachother in the same Company. Right before Pearl Harbor the National Guard was Federalized, and enlistments were extended. National Guard units, along with regular Army units would now serve and fight overseas.
The problem was regular Army Divisions were made up of people from all over the USA. National Guard units were not - until later in the war, when the draft and enlistment helped rebuild numbers.
A perfect example of how this was a problem was on D-Day, Omaha Beach, Normandy, France. The 116th Infantry Regiment of the 29th "Blue and Gray" Infantry Division was the tip of the spear at their sector of Omaha Beach. The 116th IR was comprised of mostly Virginians (where the regiment was originally from). The company to lead the assault for the 116th IR was "A" (Able) Company. Able Company was comprised almost entirely of men from the same town and county in rural southern Virginia, called Bedford.
These men grew up together, and went through years of training in Virginia and then in England, all for the invasion of France. The morning of June 6, 1944 these men were the first to land on their portion of Omaha Beach. This company was almost entirely decimated by withering German MG42, artillery and mortar fire. Percapita, this small town in middle of no where rural Virginia suffered the largest lost of life (for Americans) on D-Day. And that is why there is a random D-Day Memorial and Museum in the rural southern Virginia. These men gave it their all, and did their duty, and paid dearly for it.
So to answer your question, there is a reason it is a common TV trope. That's how units were organized in Europe and to a degree in America. And this military style of organization caused a lot of heartache and pain for small towns everywhere, when there young men would be slaughtered in droves - like the Bedford Boys on June 6th, 1944.
Read the book "The Bedford Boys" by Alex Kershaw.
Wow thanks so much