Did religious names catch on before or after the use of last names?

Short Answer:

In my humble opinion, religious and biblical names probably became common in western Europe centuries before family names became common. I do not think that people would be afraid of widespread use of a few personal names leading to confusion between different people with the same personal name, and so avoid using religious names. I think that people found ways to cope with two or more persons with the same personal name in a community and wouldn't use that as a reason to avoid using any particular type or set of names.

Longer Answer:

I note that family or clan names were used in some ancient cultures, most fmaously the Roman Empire, where the last known examples probably date fromthe 6th century AD.

Of course the eastern section of the Roman Empire, the so-called "Byzantine" Emire, continued for many centuries after that. During the Middle Ages surnames in Greek began appearing in records.

Among famous noble and sometimes imperial families, the Doukas surname may date back to the the regency of Empress Theodoro in 842-855 when a "son of Doux" is mentioned. The Kantakouzenos family and the Angelos family are first menitioned about 1100. the first mention of a Phokas family member is in 872. The first known Skleros was mentioned in 805.

So the first "Byzantine" surnams may have developed during the 8th century, about two centeries after the last ancient Roman surnames were used.

By 1400, most English and some Scottish people used surnames, but many Scottish and Welsh people did not adopt surnames until the 17th century, or later.

In England, the introduction of family names is generally attributed to the preparation of the Domesday Book in 1086,[citation needed] following the Norman conquest. Evidence indicates that surnames were first adopted among the feudal nobility and gentry, and slowly spread to other parts of society.

Surnames were uncommon prior to the 12th century, and still somewhat rare into the 13th; most European surnames were originally occupational or locational, and served to distinguish one person from another if they happened to live near one another (e.g., two different people named John could conceivably be identified as 'John Butcher' and 'John Chandler').

https://en.wikipedia.org/wiki/Surname#History[1]

During the modern era, many cultures around the world adopted family names, particularly for administrative reasons, especially during the age of European expansion and particularly since 1600. Notable examples include the Netherlands (1795–1811), Japan (1870s), Thailand (1920), and Turkey (1934). Nonetheless, their use is not universal: Icelanders, Burmese, Javanese, and many people groups in East Africa do not use family names.

https://en.wikipedia.org/wiki/Surname#Modern_era[2]

So this information on the dates of adoption of surnames in varius cultures shows that in many culture there were communities of thousands, and tens of thousands, of persons, for many centuries before surnames became universal or common in those cultures.

So in any town or city in western Europe, and in many villages, there would have been examples of two or more persons using the same personal name at the same time. And that would have gone on for centuries after Roman clan names disappeared and before family names were first used.

So people were used to multiple persons with the same personal name in the community and had ways to tell them apart for centuries before family names began to be used. So why would people avoid giving biblical personal names to their children in order to avoid duplication of personal names in the community?

When a pagan society was converted to Christianity, people with names usual in their pagan culture would often be baptised with Christian names, which were often biblical names. And perhaps the custom of having both local and baptismal names continued for generations after a society was converted to Christianity. And maybe Christian names, including biblical ones, would gradually be considered to be native names.

Monks and nuns often gave up their birth or baptismal names and adopted religious names, often biblical ones.

Therefore, I guess that many religious names were used as first names long before the use of family names became commoon in western Europe.

According to wp, in Germany, family names appeared first in the cities (from the 12th century onwards). Having no family name was still quite common in the 14th century and in the countryside family names became only necessary in the 17th or 18th century.

If we look at a few name lists, most Minnesängers from the 12th to 14th centuries seem to have had Germanic names and no family names. Among the mayors of Hamburg, there already are some "Johannes" at a time when patronymics and birthplaces still seem to be used instead of family names (e.g. Johannes filius Oseri). A list of names of people who lived around the 1525 peasant's revolt has quite a lot of "Hans" (a short form of Johannes). Although a lot of them do seem to have had some kind of surname.

So my impression is that

a) definitely religious names were common before every last peasant had a family name

b) the problem does not really go away if people do not use religious names. Just look at the number of Heinrichs in those lists

c) if we take a look beyond Europe and Christianity, family names like Hosseini or Mohammadi are really common in Persian-speaking countries. As far as I am informed, this means that some grandfather in the family was named Hossein or Mohammad at the time surnames were introduced, which in turn would mean that at least in that part of the world, religious names were common before family names became necessary.