Although I only touched on naming, especially in China, in my research on Japanese naming practices, I might be able to answer this at least sufficiently.

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For starters, we first need to ask: what is a surname?

In its most basic definition, a surname is a secondary name, used in addition to a first name (a.k.a., given name). Unlike the first name, which identifies the individual, the surname defines the individual’s membership in a social unit; typically, a kinship group: a clan, tribe, or, especially in our modern times, the family. What is important is that a surname is hereditary: it is passed from parent to child.

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(Western) Europe

But surnames did not exist since the beginning of time:

In many cultures, the surname emerged from the practice of byname (some also call this a nickname). The byname, like the surname, is a secondary name that served to further identify an individual.

After all, it’s a bit problematic when everyone only has a given name, and naming children after biblical figures (etc.) becomes all the rage—the sheer number of people named “John” in medieval English sources should be sufficient proof for this!

Either way, byname naming practices are dime a dozen and differ by culture in their variety and distribution. To give some general examples for all the many Johns out there: there might be a John, son of David (i.e., Davidson), John the Miller, John of Reddington, John from(!) Reddington, John from the Hill, John the Tall, John the Bear, and so on.

These kinds of bynames became hereditary over the course of centuries (in Western Europe, around the 11th to 14th century, give or take); that means, they stopped being the secondary name to only a single person, but were applied to their children, as well (2). This also means that these byname-into-surnames lost their original meaning—John the Bear was John the Bear for a reason! This is one of the most common patterns of surname formation, especially in Western Europe, but also observable in other parts of the world (for example, Japan). It is also quite typical that such surnames can first be observed within members of the aristocracy, for whom both genealogy and also identification of the place they ruled was of particular relevance. (1)

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China

On the other hand, in China, some form of surname has been in use since the first millennium BC. This is possibly the oldest occurrence of surnames in the entire world.

However, in China, surnames existed in the form of clan names, used by all members of the same extended family group (so, not just parents and children, but parents, all their siblings, all their children). There also was another type of surname, referring to all members of a certain territorial-political unit, which would be shared by a regional ruler (or similar figure) and all his dependents, regardless of actual biological kinship (this might be dubbed a tribal name for the sake of distinction). These two origins of surnames converged into the singular concept of clan name xìng 姓 by ca. 200 BCE. This, the Chinese surname, did not change on marriage, since it was directly tied to ancestry. It was transmitted from father to children. Incidentally, marriage between people of the same surname was forbidden; adoption of a child of another surname was forbidden. This old clan name-type of surname was, at least according to my sources, only really replaced with a modern idea of surname by law in the People's Republic of China in 1950 (3).

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Conclusion

As you can see, the surnames in use today have entirely different roots, of which the origin in the byname provides for a much wider spectrum of possible names. Bynames originated as descriptors for individuals, whereas clan names in the Chinese tradition referred to larger groups, therefore inherently limiting the number of names emerging. I should also note that there are other types of name formation as well, not just the Chinese clan-name/surname and the byname/surname.

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Incidentally, this Chinese idea of surname (clan name) was also introduced to other Asian countries in the Chinese cultural sphere, such as Korea and Japan (the latter adopted it ca. in the 8th century). In East Asia today, there are about 3.000 known Chinese, only about 200 Korean, but over 100.000 Japanese surnames in use. This is because in Japan, the Chinese concept of clan name was displaced over the course of the 12th to 16th century by the emergence of bynames, which became hereditary surnames (hmm...where have I heard this before?) (4).

Furthermore, the U.S. are probably a bit of anomaly, due to their history as an immigrant country, which means that surnames from numerous (especially Western European) origins were pooled together, which should facilitate extreme diversity of surnames. (I'm 100% certain someone did studies along the lines of surname distribution and cultural origins in the U.S.)

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Hope this was informative enough to be sufficiently answering your question ^^

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Notes:

(1) The first paragraphs synthesize fragments of information from various chapters in the 2016 Oxford Handbook of Names and Naming, edited by Carole Hough, and a tiny bit from essays in 2007’s Prosopography Approaches and Applications: A Handbook, edited by K. S. B. Keats-Rohan.

(2) It goes without saying that the descriptor separating the given and the byname were eventually omitted for convenience.

(3) Ōtō Osamu. Nihonjin no sei, myōji, namae: Jinmei ni kizamareta rekishi. Yoshikawa Kōbunkan: Tokyo, 2012, pg. 8–12.

(4) Sakata Satoshi. Myōji to namae no rekishi. Yoshikawa Kōbunkan: Tokyo, 2006, pg. 3–4.