Before 1871, Japan had two kinds of what can be called family names: the uji and the myōji. The uji was a classical surname recognized or given by the Imperial Court. It was more similar to Chinese and Korean surnames, and the number of uji was fairly limited. Though Shinsen Shōjiroku (815) lists 1,182 prominent uji in the capital region, many of the Imperial positions were held by Fujiwaras in the 8th century.

In the 12th century, when the authority of the Imperial Court declined and samurai clans emerged as local leaders, those clans, mostly Minamoto or Taira, started to distinguish each other mostly by locations, which later developed into the myōji. Without regulation by the central government, they created and changed myōji at will, when they moved to a different location, developed a new land, or established a branch clan.

In the Muromachi period, the practice of myōji extended to lower social classes. Their myōji were often named after small toponyms or geographical features only understood within each village, which created countless myōji whose origins are now unclear.

The uji was still used in official documents until the Edo period, like Taira by Hōjō, Oda, Minamoto by Ashikaga, Tokugawa, etc., but it became only ceremonial, and claims made after the Sengoku period are highly questionable.

After the Imperial Court regained authority in 1867, they initially addressed their officials with uji in official documents, but it expectedly didn’t work. They abolished the use of uji in 1871, and ordered all citizens to officially use myōji in 1875. On this occasion, a small percentage of families registered surnames of their own creation, which added up the total number of surnames in Japan.

Nobody mentioned Korean last names. I will add up just a little more about that for your information, even if it's 5 days ago. The book 택리지 擇里志 written in 1751 Joseon dynasty by 이중환 李重煥, says that Koryeo adopted chinese family name system to the people. Koryeo is the dynasty which united three-divided nations in 936 before Joseon. Before that, there were already some last names. For example, ancient kingdom Silla was rulled by three royal families, which are 김 金 Kim, 박 朴 Park, 석 昔 Seok. All of these were written in chinese letter, because there was not korean own alphabet then. There were also chinese immigrants who used their chinese family name in korea. Such as 제갈 諸葛 독고 獨孤 황보 皇甫 are known as originated from chinese ancestor. And there were 노비 奴婢 who did not have family name until late Joeseon. Many of them bought 족보 族譜, a book of family tree, to become one of its family and finally got family name. It seems the reason why there are so many Kim in modern korea causing lack of diversity in family names.

When you say “last name” do you mean family name or their personal names, since East Asian naming conventions typically have the family name first.