The name Hàn (汉) is not a common Chinese surname despite it being the name of the Han Dynasty. It is not in the list of top 100 most common Chinese surnames. The much more common 'Han' is Hán (韩) which is prevalent both in Korea and China. Hán (韩) was a state in the Warring States period that was ultimately conquered by the state of Qin in their unification of China. Hán (韩国) is also the name for South Korea (North Korea sticks to the name Chaoxian (朝鲜) from the Joseon/Choson Dynasty).

Out of all the major Chinese dynasties, Zhou (周), Tang (唐), and Song (宋) are the Chinese dynasty names that are also common Chinese surnames. The imperial surnames of every Chinese dynasty after the Qin Dynasty (except the Yuan and the Qing who were not sinicized at the beginning of their rule) are also common surnames today. Liu (刘) was the imperial family surname in the Han Dynasty, Li (李) during the Tang Dynasty, Zhao (赵) during the Song Dynasty, and Zhu (朱) during the Ming Dynasty. I don't know how common each surname was before their respective dynasties came to power. Commoners did not really use surnames until the Qin Dynasty. When choosing a surname, they often would take the name of their state or fiefdom (for example Han (韩) and Zhao (赵) are both very common surnames and states conquered in the Qin unification of China). Origin of Chinese surnames.

Examples of adoption of Chinese surnames as a part of sinicization: Xianbei changing their surnames to Chinese surnames and Muslims adopting Chinese surnames.

With regards to Arabic, they of course historically did not follow western naming conventions. It breaks down as follows:

The اسم or Ism, the personal name, like Ali, or Muhammad. They had rather simple meanings, just like Peter comes from Greek πέτρος, which means stone.

The كنية‎ or Kunyah, means "Father of", and is usually the name of your oldest child, signified by abu or umm. Note this was not always a hard rule however, as in the case of Caliph Abu Bakr. Bakr means a young camel, and he was given the name because he loved camels as a youth.

The نسب or Nasab, which details your patrilineal descent, signified by ibn for men or bint for women (Joe, son of Bill, son of John, son of Peter, etc; or Susan, daughter of Joe, son of Bill etc).

The لقب or Laqab, which works as a sort of personal title or nickname. "Rashid", or more accurately "Al-Rashid" (which means "the Guide") is a Laqab, and was not used historically as a surname or the name of a dynasty.

The النسبة or Nisbah, which is a marker showing descent from a clan or location, signified by the suffix -iyy for men or -iyya for women. Al-Qurayshi for example means "Of the Qurayshi tribe", or Al-Tikriti means "of Tikrit".

Take for example the first Caliph of the Abbasids, known in English as simply As-Saffah. His full name was Abu al-‘Abbās ‘Abdu'llāh ibn Muhammad as-Saffāḥ. This means Father of Abbas* (Kunyah), servant of God (Ism), son of Muhammad (Nasab), the Generous (Laqab). His given name was Abdullah (servant of God).

*Abbas can mean stern, grim, to frown, to scowl, or a term for a lion.

The Nasab could be truncated to just your father or further extended when you wanted to.

So you can see there never really existed such thing as a dynastic surname in Arabic for people to copy - the Nasab, and later the Nisbah were much more important with regards to proving descent. Even today, the Hashemite King of Jordan usually goes by Abdullah II bin Al-Hussein - Abdullah II, the son of Hussein. His father was Hussein bin Talal - Hussein, the son of Talal.

I should also mention two titles: شريف‎ or Sharif, and سيد‎ or Sayyid. They are used to show descent from the grandchildren of Muhammad. Sharif was used for descendants of Hasan ibn Ali, and Sayyid was used for descendants of Husayn ibn Ali. Note again these are inherited titles, not part of one's actual name per se.