"A historical outline: the earliest official registration of an individual’s personal details in the West dates back to the time of Marcus Aurelius, who made it obligatory for all fathers to register the birth of all children. This practice continued until the fall of the Roman Empire, when it lapsed, but was reintroduced in 1563. From that moment we owe the most precise records to the Church, before the introduction of the Civil State Registry Office in 1861 after the reunification of Italy." http://www.youritalianorigins.com/en/services/genealogical-research/
Some believe that "Marcus Aurelius required a birth registration. He was the f irst emperor to do so." http://department.monm.edu/classics/Courses/CLAS201/ChildhoodinRome.pdf
This is likely due to the Historia Augusta which is not considered entirely reliable.
It is almost universally accepted that Augustus began official government registration of births. [https://books.google.com/books?id=iTpa6IH3XHMC&pg=PA179&lpg=PA179&dq=Marcus+Aurelius+%22birth+registration%22&source=bl&ots=sYELGQBx-Q&sig=8iNDcuEXCoyNGb4TsxHoKl5FNFE&hl=en&sa=X&ei=-MreVKHDMs23ogSv04HoDA&ved=0CCkQ6AEwAg#v=onepage&q=Marcus%20Aurelius%20%22birth%20registration%22&f=false Old Age in the Roman World: A Cultural and Social History
By Tim G. Parkin]
It is believed that Augustus offered registration of birth to Roman citizens who wanted to benefit from his new deal after the roman civil war that brought him to power and brought the Republic to its knees.
It was Marcus Aurelius that made birth registration manditory. [https://books.google.com/books?id=vh3pmAodawEC&pg=PA88&lpg=PA88&dq=Marcus+Aurelius+%22birth+registration%22&source=bl&ots=yEXjKoJwVJ&sig=irXhuLLqmVYTfbbJpUTBpcq_HWE&hl=en&sa=X&ei=-MreVKHDMs23ogSv04HoDA&ved=0CDoQ6AEwBg#v=onepage&q=Marcus%20Aurelius%20%22birth%20registration%22&f=false Debating Roman Demography
edited by Walter Scheidel]
This was only the first step in a three part process where the state would make a son a member of a civil family. It was the sacrifice of the son to the temples upon receiving his toga that finalized the process. [https://books.google.com/books?id=JbpEAAAAQBAJ&pg=PA58&lpg=PA58&dq=Marcus+Aurelius+%22birth+registration%22&source=bl&ots=LpM4aibc24&sig=RPzBIP0xUlGLngLDOLhrkJY2TZk&hl=en&sa=X&ei=-MreVKHDMs23ogSv04HoDA&ved=0CD4Q6AEwCA#v=onepage&q=Marcus%20Aurelius%20%22birth%20registration%22&f=false The History and Theory of Children’s Citizenship in Contemporary Societies
By Brian Milne]
"Having a child was not something Roman women looked froward to. Many women died in childbirth, or miscarriages. If the birth was successful, the father would take the child in his arms to show that he was the father. Nine days after a boys birth, he would be given his first name and his bulla, a good luck charm that warded off evil spirits. This would happen for girls eight days after birth. During the rein of Marcus Aurelius, birth registration was required. The parents of the child had to register the birth at the public records office within 30 days. THis custom has survived till modern day." http://igonzales.weebly.com/childhood.html
These laws concerning the manumission of a son from his his natural father was a transfer of the Parens Potestas of the father to the state. The word Manumission is from manumit. It was normally considered "the act of a slave owner freeing his or her slaves. Different approaches developed, each specific to the time and place of a society's slave system. The motivations of slave owners in manumitting slaves were complex and varied."
This manumission was done formally by records of the birth but informally bu conditions that allowed an assumption by the courts or as a favor of political powers under Justinian. Borkowski's Textbook on Roman Law, By Paul du Plessis
"These ceremonies seem to have been private, that is, it cannot be shown that there was at this time any taking of the child to a templum, as there was among the Jews, or any enrollment of the name upon an official list. Birth registration, which many of our own states have been slow to enforce, was first required under Marcus Aurelius, when it was ordained that the father must register the date of birth and the name of his child within thirty days, at Rome before the praefectus aerāriī, in the provinces before the tabulāriī pūblicī. In the case of the boy the registering of the name on the list of citizens may have occurred at the time he put on the toga virīlis." 
"For example, the detailed aspects of the birth records indicates that somebody gave significant thought to the process. The birth registration had a master copy on a wooden tablet (album) stored by month/year in the Forum of Augustus (or provincial office/Tabularium), a codex copy of the contents was made, each document had an 'quasi-abstract' on the outside of the document and the relevant party had their own copy. " 
- "A certain number of such boards formed a unit. We may call these units for convenience sake volumes, although the boards were not bound together. Each board in such a volume had a certain number and was divided into several columns which were called paginae. There existed, however, at the Tabularium besides the album another register of births of a more handy form, written in a codex or a papyrus roll. Such a book was absolutely necessary for two reasons. It was technically almost impossible to record at once every professio on the large wooden boards of the album. The official certainly first took the profession down ad acta, and later, when several professiones had come together, ordered a painter to make the entries in the album. On the other hand the album was not kept for an indefinite time. According to Roman administrative custom an album had only a short life. It was indeed quite impossible to store these alba for years and years, for very soon a vast forest of such boards would have grown up. The album was removed and destroyed after a certain time, only the register written in a codex or a papyrus roll was preserved at the Tabularium. This book, being written in the form of a diary, was called kalendarium. … So the Roman register of births consisted in fact of two registers, the kalendarium and the album, both of course in substantial agreement with each other. Each professio was first recorded in the kalendarium and afterwards, ad kalendarium (according to the kalendarium), in the album." ["Roman Registers of Births and Birth Certificates", Fritz Schulz, Journal of Roman Studies / Volume 32 / Issue 1-2 / November 1942, pp 88; DOI: 10.2307/296462, Published online: 24 September 2012]
And, although the full procedure was only available to full citizens, there was a substitute allowed for non-citizens or 'half-citizens' ('illegitimate'):
- "This registration, however, as established by the two Augustan statutes, was strictly confined to legitimate children in possession of the Roman citizenship. (a) The register was barred to illegitimate children. This is expressly stated in no. 16. There was, however, a substitute for the forbidden registration. The parents were allowed to make a private statement of the birth before witnesses, a so-called testatio and to draw up a document on this declaration. Whether such testationes were expressly allowed by the two Augustan statutes, is as yet not quite clear. Reading no. 15 (' se testari ex lege Aelia Sentia et Papia Poppaea ') one might be inclined to believe it. Anyhow they were not forbidden, as our documents give us examples of them (nos. 12-16)." [Schulz, op. cit. 81]