It depends on who you mean by “we”. Family names are a cultural institution, and different cultures vary widely as to when they adopted their use.
The Chinese began using family names during the reign of the Emperor Chin, who built the Great Wall of China, and from whose name the word “China” is derived. He commanded that families choose a word from a poem which was a favorite of his as their family name which, in Chinese practice, is the first name; hence one spoke of “Chairman Mao”, not “Chairman Tung”. The other words used to make up an individual name also were to come from this poem.
An interesting consequence of this is that all Han names (The Han being the principle ethnic group in China) has a literal meaning; Chairman Mao’s name meant “Hair Enriches the East”. Another consequence is that all family names are likewise personal names; for comparison, imagine if all English people had last names such as Sheldon or Bennett. And there are only about 200 last names in use, with the names giving very little suggestion as to where a family comes from originally, or to whom they might be related.
As noted earlier, the Turks did not adopt last names until the 1930s. This was part of Ataturk’s efforts to westernize the nation. Ataturk, incidentally, invented his own last name.
Jews from central Europe largely acquired their family names as a consequence of censuses taken in late Renaissance (to use a word I can never remember how to spell). Mark Twain discussed this in one of his essays about anti-Semitism, noting that a number of Jewsih families got stuck with monikers which seemed rude or unpleasant as the result of dealing with unsympathetic census officials.
Irish names generally refer to lineage traced back to the eleventh or twelfth century or thereabouts. People named O’Brien, for instance, claim descent from Brian Boru, the first king of a united Ireland. This is not much of a distinction considering the time that has elapsed; a geneologist hired by Ronald Reagan determined that he was related to the current queen of England because they could claim Brien as a common ancestor.
In some cultures people still get by without family names. In Iceland, for instance, a man’s last name is based on his father’s first name. Thus Lief Bjornson may have a son named Bjorn Liefson, who has a son named Lief Bjornson, who has a son named Bjorn Liefson…