Patronymic surname should not be confused with Patrilineality.
A patronymic surname is a surname originated from the given name of the father or a patrilineal ancestor. Different cultures have different ways of producing patronymic surnames.
In the Old Testament of the Bible, men are identified by their lineage through use of their father's first (and only) name.
Last names were ‘normalized’ and became more standardized with the advent of mass literacy, paper availability and documentation, and mobility. For example, passports vs early letters of introduction for travel.
For example, early patronymic Welsh surnames were the result of the Anglicizing of the historical Welsh naming system, which sometimes had included references to several generations: e.g., Llywelyn ap Gruffydd ap Morgan (Llywelyn son of Gruffydd son of Morgan), and which gave rise to the quip, "as long as a Welshman's pedigree."
As an example of Anglicization, the name Llywelyn ap Gruffydd was turned into Llywelyn Gruffydds; i.e., the "ap" meaning "son of" was replaced by the genitive suffix "-s", but there are other cases like "ab Evan" being turned into "Bevan". Some Welsh surnames, such as John or Howell, did not acquire the suffix "-s." In some other cases the suffix was affixed to the surname much later, in the 18th or 19th century.[1] Likewise, in some cases the "ap" coalesced into the name in some form, as in Broderick (ab Rhydderch), Price (ap Rhys) and Upjohn (ap John).[2] [3]
Similarly, last names or surnames were not set in Russia, but patronymic and based on a father's first name.
Peter's children might be Jan and Roman Petrovich. Peter's grand children could be Aleksandr Janowicz and Ivan Romanowicz/ Romanovich. Peter's great grandchildren could be Nicolai Aleksandrovich and Dmitri Ivanovich.
Nearly all Icelandic surnames are strictly patronymic, or in some cases matronymic, as Icelandic society generally does not make use of hereditary family names. A similar situation could, until relatively recently, be found in the continental Scandinavian countries of Denmark, Norway, and Sweden, where hereditary family names came into widespread use for the general population during the course of the 19th century or the beginning of the 20th century, ultimately as a consequence of legislation. As the outcome of this, a large majority of Scandinavian family names originated as the patronymics borne by the heads of family at the times when these laws came into effect, and these surnames mostly display a limited variety reflecting the popularity of male given names during the 18th and 19th centuries in those countries.
Heraldry, like early pictographs, logos or icons, also helped designate families across much of Europe. People working for a landowner, or living in an area might adopt the same crest, especially when landowners had to provide troops to their nation or king for battle.
Scottish clans and tartans served as family or area identifiers.
Of note, some Asian countries list family name first, most adapt to given plus family standard when abroad. Whereas, some Latino countries add mother's family name at the end.