Comparative diversity of European family names
In many cultures, surnames meaning "son of ..." are predominant. This is the case of Scandinavia, Russia, Northern England (of Scandinavian origin), Spain, and to a lesser extent also Ireland and Scotland.
The English, French, Dutch, German and Italian speaking cultures have several categories of names, including the above, but also geographic place names, village, town or region's names, jobs, physical descriptions, or names with no meaning (particularily common in French and Italian).
France probably has the greatest variety of surnames of any country in the world : about 900,000, including foreign names. Italy comes second with 350,000. But it is Belgium that has the greatest number per inhabitant, with 190,000 for a population 6x smaller than France or Italy. Italian surnames are spread out more evenly, and consequently have the smallest percentage of population represented in the 100 most common names.
If we exclude very rare surnames, name diversity can be roughly assessed by making the total percentage of people carrying the 10 (or 50, or 100) most common surnames in the country.
10 most common surnames by country (percentage of total population)
Belgium : 2.03% (top 50 = 6.06% ; top 100 = about 9%)
Denmark : 25.93%
England & Wales : 5.7% (top 50 = 14.55% ; top 100 = 20.78% ; top 500 = 39.33%)
Finland : 3.87%
France : 1.89% (top 50 = about 6% ; top 100 = about 9%)
Germany : 4.09%
Italy : 0.67% (top 50 = 1.76%, top 100 = 2.55%)
Netherlands : 3.99% (1947 data)
Norway : 8.75%
Poland : 2.87%
Spain : 19.65%
Sweden : 19.5%
Scandinavian countries and Spain clearly have fewer surnames, with about 1/4 to 1/5 of the people sharing the same 10 surnames only, against only 0.67% in Italy, about 2% in Belgium and France, and 4% in the Netherlands, Germany and Finland.
Comparison with a few Asian countries
Japan : 8.35% (top 50 = 19.7% ; top 100 = 28%)
Vietnam : 82.9% (!!!)
China : (the top 3 names cover 22.4% ; top 19 = 55.6% ; top 100 = 87%)