In feudal Japan, retainers and subjects would often adopt the noble name of their overlord, so there would be several people within a clan who held the same last name but were genetically unrelated.

However, during the Meiji restoration, the government required men to choose new surnames in order to break up clan distinctions. This happened at a time when the population was in the millions, so nowadays there are over 100,000 surnames still in use in Japan. So most people with the same surname today probably have the same common ancestor, but from that point in time and not earlier, from clans.

Looking through old saved threads, and found this one I never got to. My apologies. The answer is mostly no. The most commonly shared Japanese surnames are no indication of a common ancestry. As mentioned by the other answer you got, surnames were formalized in Japan very late, in the 1870s, with a large part of the country either not having surnames before that, or more likely, bearing unofficial surnames unregulated by government.

However, the other reply is wrong in thinking that names in common mean a common post-Meiji ancestry. The most common surnames in Japan are place descriptors, denoting where the family lived. So "Yamamoto" = "base of the mountain", "Tanaka" = "middle of the rice paddies etc. The same names, therefore, could be used in completely different areas of the country, by unrelated families to describe their location.

I wrote a longer reply on the history of Japanese surnames in general here.

If not with the Japanese is there any group for which this is the case?

You might want to ask another question about the history of Korean surnames, clans, and how shared ancestry is denoted. I can't really tell you about them myself, except that it's a fascinating subject.