It definitely differs by ethnicity. An example is African Americans. When slavery ended, former slaves picked their own surnames. Washington and Jefferson were two popular choices. Of course, this means that not all Washingtons inherited their name from a common ancestors.

Also, sometimes (though not as often as commonly thought) former slaves took the surname of their former masters, or the master of one of their ancestors. (An example from California history) This also means that the descendants of the former slave and the descendants of the former master may share a name without sharing an ancestor.

Edit: as a bonus, here's a NY Times article about why many black Americans have Irish surnames.

Geneticists at the University of Leicester in the UK did a series of studies on this a few years ago. Here's a quote from their webpage

What did we find?

We found that men with rare surnames - such as Grewcock, Wadsworth, Ketley and Ravenscroft - tend to share Y chromosomes that are very similar, suggesting a common ancestor within the past 700 years. However, men with frequent surnames, such as Smith, are no more likely to have a recent common ancestor than men chosen at random from the general population - they derive from a person's trade and would have been adopted many times by unrelated people.

Does it differ by nationality or ethnicity?

I would say it certainly does. Take a look, for example, to Latin Americans, more specifically Hispanic Latin Americans. Most of them bear Spanish surnames, mostly of Castilian origin (Pérez, Fernández, Hernández, Suárez, etc.), but also of Galician, Catalan and Basque origin (Echevarría, Iturri, Goyeneche, ...). It is not infrequent to find indigenous people of South and Central America with these surnames as well.

However, the cause of this is, in most cases, not a single common ancestor. During colonial times, the Spaniards took it upon themselves to Christianize the indigenous people of the Americas. They would baptise them and give them new "Christian" names and surnames. These would be names that were common in Spain (Juan, Pedro, José, María,...) and, as well, common last names (Hernández, Pérez, Flores, Puente, Castillo, etc.).

A revelent part of the Hispanic population in the Americas has Basque surnames. Yet, not many of them are of Basque origin or have Basque ancestry (some of them do, of course, but they are a minority). We can assume the same happens with Castilian and Catalan surnames.

In the case of Korea, there are many "Clans" (called Bon-gwan) with the same family name, but with a different "origin".

Thus two people with the same family name "Kim" may not necessarily be descended from the same ancestor, as one could be from the Gyeongju Kims, and the other could be from the Gimhae Kims.

However, those from the same "Clan" will (in theory) have inherited their family name from the same distant ancestor.