In China, the top 3 surnames have a population larger than that of Indonesia. In Vietnam, roughly 40% of the population has the last name Nguyen.
However, this is much less the case in most western cultures. Smith, Jones and Williams were the most common last names in England in 1991, but they only add up to be 2.75% of the total population.. In Spain, the numbers are higher, but they are still nowhere near the prevalence of the Asian surnames..
First, they don't have so few. The common ones are just very common. About a decade ago I started collecting a database of Chinese surnames. I eventually stopped because it was getting ridiculous. I had close to a thousand when I decided to bail on it because I saw no end. Yeah, a lot of people are named Bai (白) or Wang (王) or Lee (李) or Park (朴) or Nguyễn (阮), but there are a huge number of other less common names.
Second, to the extent they do, it's because people are pulling from a much more limited source of traditions. It's unreasonable to compare surnames in immigrant-heavy cultures like the US or EU (where people are coming from a wide range of linguistic backgrounds) versus the Sinosphere where people are actively adopting Sinitic surnames.
Vietnamese surnames are generally Sinitic. Nguyễn, Trần, Lê, Phạm, all Sinitic. The vast majority of Korean surnames (by population) are generally Sinitic. Between Korean and Vietnamese, we've only got a few hundred names, and a lot of those exist in both. Certain surnames like 金 (Kim) or 南宫 (Namgung) are more common among but not limited to Koreans. 阮 (Nguyễn) is more common in Vietnam but not at all limited to Vietnam.
If you look at the history of Chinese surnames, a common trend is people either being given a surname by the state, or adopting a surname to show loyalty to the state. Surnames changed all the time, historically. The article you linked to on Nguyễn gives a good overview of exactly that.
But again, we need to compare apples to apples and look at the reasons surnames are so varied by comparison in places like America. You have immigrants, so you have more names coming in. Around 75% of Korean surnames are not Sinitic now. There are around 6000 surnames registered in South Korea today.
And it's still happening. The vast majority of ethnic Chinese in places like Thailand now have Thai surnames.
Add to that the fact that surnames in Europe are sometimes a lot more recent. In China, family lineages started being taken incredibly seriously all the way back in the Song period (1200s). A lot of European surnames are much more recent than that, even if the family histories are not. The Netherlands didn't standardise or require surnames until quite recently, for example. Additionally, you could be named for your occupation, your hometown, a geographical feature your family lived by, and so on. Since the Netherlands didn't require surnames until 1811, there hasn't really been much time for those to select out as has been the case in Asia. Not every Trump is going to have male sons forever (in part since Trump itself is a new surname) so we can expect that in a hundred years, there may not be any more Trump-named people. Same for Disney, again a somewhat new surname (though d'Isigny will more likely survive longer).
A lot of people also changed their own surnames to new names when immigrating to places like America. That also ups the numbers, though I couldn't give you hard statistics on that.
tl;dr: Immigration from more traditions and younger traditions means means more names. But don't underestimate just how many surnamed there really are, even in the Sinosphere.
In the case of Vietnam, as I'm sure is the case in China & Korea, the popularity of last names is extremely lopsided. For Vietnam, a survey was conducted by the University of Social Sciences and Humanities (Ho Chi Minh City) of 805 Southerners and 1,136 Northerners found that, amongst them, the 3 most popular last names were in line with the government stats. In all of Vietnam, 38.4% have the surname Nguyễn, 11% are Trần, 9.5% are Lê a.s.o. These 3 last names alone account for a small majority. What you'll immediate find is that all 3 of these surnames were that of imperial dynasties in Vietnam's history.
In 1232 the Trần prince Thủ Độ wished to erase the influence of the Lý Dynasty and forced people to adopt the surname Nguyễn citing punishment for those who didn't.
Emperor Hồ Quý Ly who overthrew the Trần Dynasty committed atrocities against members of their imperial family, also convicting about 300 people for plotting to revive the Trần Dynasty. After his dynasty fell in the year 1407 he ordered his family to adopt the surname Nguyễn.
The collapse of the Mạc Dynasty in 1592 saw members of that imperial family adopt Nguyễn and Lều as last names.
After the Trịnh-Nguyễn Wars, many members of the Trịnh family adopted the enemy's last name for fear of retribution (this is a recurring theme).
Many criminals during the 1800s would adopt new last names.
The reasons above account for why there are so many people with the last name Nguyễn and a handful of others (mostly from imperial dynasties).
Người Trung Quốc có 926 họ, người Hàn Quốc có 274 họ, người Anh có khoảng 16.000 họ, người Nhật dẫn đầu thế giới 100.000 họ. Còn việc xác định có bao nhiêu dòng họ ở Việt Nam là một công việc khó khăn. Nhiều con số khác nhau đã được đưa ra 202, 300, 191.
"The Chinese have 926 last names, the Koreans have 274, the English have about 16,000, the Japanese lead the way with 100,000 last names. As for Vietnam it's a difficult job to specify how many there are. Many different figures have been giving such as 202, 300, 191."
There are 264 or so family records dating back to 1533 kept by the Hán Nôm Research Institute. The bottom line is that Vietnam's last names are mostly derived from Chinese (the whole dynastic setup was installed in a similar fashion as China's, the official written language was Classical Chinese). Last names in Modern Vietnam are remnants of the dynastic past, depicting the clan mentality. Whereas in Europe, especially in England, where last names come from a vast array of backgrounds (professions, place names, descriptions, a.s.o) those in Vietnamese are relegated to a fixed number of clan names, many of which are quite arbitrary.
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