This is a very broad question, and a lot of it has to do with what type of unit / specialization, which country, when etc...

Although I am unfamiliar with the show, I have read the book by Kershaw. The book centers around a soldier, Felix Sparks, who is a soldier in the 45th Infantry Division, during its liberation of Europe from Sicily / Italy to Germany. The unit in question was originally an Oklahoma / and Texas National Guard division.

During World War II (as well as before and after) the US Army National Guard was an individual state, or region's state level Army, unlike the regular Army which was on a national level, and had divisions which were compromised of soldiers from all over the USA and its territories. The regular Army's initial goal was to defend and fight for the country as a whole. The National Guard were state's organized militia / Armies. They were not comprised of regular Army full time soldiers from all over the country, but rather, soldiers with civilians jobs and lives who would get called up for training once a year, and or during local emergencies (such as natural disasters, or large rowdy protests). A lot of men joined the guard before the War as a way to earn some extra money during the depression, or in some cases, as something to do and as a kind of surrogate good ol' boys club.

The peacetime guard, and overall Army in general, was reduced from its WWI wartime strength, and promotions and changes moved slow as molasses. It was very common in many of the guard units to be with the same group of guys you grew up with, as most guard Divisions were by state / region, regiments by state / part of the state, and companies were made up from individual towns and part of the state. People stayed for a while, as it was relatively easy money during the depression. It was not uncommon to be a private for several years, and have NCOs and low level officers who were a bit older. But again, these were men who had lives outside of their weekend soldiering.

Then in 1941 FDR federalized the National Guard, and started the draft. So once weekend warriors now became regular soldiers. Once only local crisis units now became regular Army units. So at the start of the US's involvement in World War II most National Guard divisions were still predominantly made up of local men, and parceled out in to companies by town and city and such. So yes, many units had men who had been lifelong friends, and many units had brothers, fathers and sons and so on. A good example of how this sort of grouping of men could become catastrophic was on June 6th, 1944 when one of the Divisions to spearhead the American part of the invasion was the 29th "Blue and Gray" Infantry Division from Virginia, DC, and Maryland, landing on the left flank of Omaha Beach (if you have seen Saving Private Ryan they are the guys with the Blue and Gray Yingyang on their helmets, as opposed to the Ranger Diamond).

The 29th ID was a National Guard Division from the Mid Atlantic, with 116th Infantry Regiment (making up the Gray part of the Yingyang) coming from Virginia. These men had actually trained for years, with mixed results, for the invasion of France. However, unlike all the other regular US Army units that were to attack Normandy that day (1st and 4th ID, Army Rangers, 82nd and 101st Airborne Divisions) the 29th was originally a guard unit, and its regiments and companies were compromised of men from the same state and town and so on. The literal tip of the spear of the invasion was the 116th IR's A Company from Bedford, Virginia (and surrounding central VA towns). This was a tiny town of a few thousand people, yet A Co. was made up mostly of men from this town. They ended up getting slaughtered on D-Day, with almost 20 men (men who knew eachother and grew up with eachother) from this town dying, and more wounded, under withering German MG and mortar fire. Per capita that little town in rural Virginia lost more men than anywhere else in the USA, and that's why there is a random D-Day Memorial and Museum in rural central Virginia. Their whole story and saga can be read in the book "The Bedford Boys".

However, as the war progressed, and formerly National Guard Divisions had been battling their way across enemy territory, and the more causalities they suffered from the original local guardsmen, their ranks were swelled with drafted and volunteer replacements from all over the USA. The dynamic and localness of these types of units would change the longer they were in the field and the more casualties they suffered. Actually, one of the survivors of A Company who was a newish replacement before D-Day, was a Jewish man from upstate New York, in a unit of mostly Virginia countryfolk. But certainly after D-Day (at least for the 29th), and similar bloodlettings of other guard units, would their casualties be replaced with new green rookies from all over.

That's how things worked in the US Army during WWII, but the same thing played out in the British military and German military to name a few. So no, these sort of scenarios were not uncommon at all. What would be more uncommon is if siblings or fathers and sons were in the same regular Army company, let alone specialized units like Paratroopers or Rangers.