Occupational surnames like Smith, Cooper, and Taylor tend to be very common in western European languages, as are patronyms like Jones -- the Welsh version of "son of John."

To use the Smith example, you have Schmidt and variations like Schmid and Schmitz (German); Smit and De Smet (Dutch), and Smed in the Scandinavian tongues.

Johnson is even more common than Jones in America.

In both examples, these names received numerical boosts via two main processes: immigration, and emancipation.

Something like six million Germans immigrated to America prior to World War 1, which marked a sea change in how non-German Americans regarded them. Up to that point, it was mainly only the Catholic German who were discriminated against severely, but once America started sending Doughboys to the front, all Germans were regarded as suspect. Frankfurters were successfully renamed hot dogs; a drive to rebrand hamburgers as "Salisbury steaks" did not fare as well. At the same time, many an American Schmidt became a Smith. (And there were a lot of them -- Schmidt is almost as common a German surname as Smith is in English.) Other German occupational names likewise were anglicized: Fischers became Fishers and Müllers became Millers.

Germans seldom use patronyms like Johnson -- they favor occupational names and first names as last names, such as Hermann, Walther, and Gunther. Nevertheless, Johnson (and perhaps Jones, too) received a boost from immigrants to America, as many Scandinavians Anglicized names like Jonsson because they proved needlessly difficult for Americans to spell. The same process applied to speakers of more exotic tongues, like Greek -- it was far easier to assimilate as "Ted Johnson" than it would have been as Theodoros Ioannis.

The end of slavery meant thousands of freedmen and women were in the market for surnames, as most were known by first names only while they were in bondage. Because they lived in the South, where immigration from continental Europe was rare (save for in French and Spanish Louisiana), they were exposed almost solely to English, Scottish, Welsh, and Irish surnames, and since names like Smith, Johnson, Jones, Davis, and Williams were common in the wider population, it was only natural than many African Americans would choose those same names. Of the most numerous African-American surnames, Pierre, the first of non-British Isles origin comes in at number 235, and setting aside names of more recent origin, like Muhammad and Ali, most of the other surnames in the top 2000 are also of French Louisiana origin: Broussard, Breaux, Thibodeaux, Guillory, Celestin, Narcisse, Guidry, Fontenot, Cormier and Arcenaux, etc).

Others chose names of famous men, thus presidential surnames like Washington (18th in frequency among African-Americans, 138th overall), Jefferson (106th-594th), Madison (443rd-1209th), Monroe (279th-567th), and Jackson (6th-18th) were often selected. Though oddly not so many opted for Lincoln. Likely such an act would have been seen as provocative in the South of the Reconstruction era.

And some seem to have been inspired by the Bible: hence the greater popularity of names like Jordan, Moses, Abraham, and Joseph.

And now I am rambling. In short, common British names were made more common by the fact that the slaves were robbed of their languages, unlike European immigrants, and some of those European immigrants chose to Anglicize their own names once they came to America, and so the most common names tended to become still more common.