There is no one reason for why or how some vampires in the series have the name that they are introduced by. Some changed their names over time or adopted a nickname as their alias. Others have kept the same first and last name or only identified themselves by their first name.
In the case of Acillius, he kept his first name from his mortal life, and never referred to himself by a surname. His name did not evolve over time or change. Being the only person named Acillius who works for The Council, he did not need any other identifier.
Caroline's name changed throughout her life. She changed her name from Isabella to Carolus after she was turned. Later, her name evolved into Caroline.
Artemis adopted the name of the Greek Goddess after she was turned. The name was chosen because of its symbology. She, like Caroline, chose names of meaning to mark the departure from their mortality.
Halifax was first known as Waleran. After he was turned, he reestablished himself as Lord Halyfax and settled in the area that would also adopt his name.
Gerard Silva slightly modified his given name from Gerardo de Silva to align himself in modern times with the Tenet of Propriety.
The Lennox family and those in their nest did not change their names throughout time. However, Koen, more commonly known as Chase, adopted his nickname as his common name before he was brought across. Stephanie and John dropped their last names after coming across. Stephanie's last name is not known. John's last name was once Turner.
Many other characters in the series will be introduced with names that have changed over time. In the future, some characters may only refer to themselves by any name they choose when introduced in the story. Usually, the older the vampire, the more fluid their name is.