It’s not a patronymic structure. “-bert” is a common element of Germanic names from “beraht,” meaning “bright.” Anglo-Saxon names were often composed of two name elements that were words in their own right, but put together were not necessarily meant to have any particular meaning.
“Robert,” “Rupert,” “Ruprecht” – they’re actually the same name.
Of course, any name–regardless of its origin–can become a patronymic-based family name, even if it doesn’t have the usual endings that mark it as a patronymic-origin name.
Yes. “Pater” = father. “Nym” = name. Any family name originally adopted to identify a person as the descendant of another person is a patronymic-origin name.
“True” patronymics, however, are relatively rare in modern society. They are used in Iceland and in certain parts of India, but are not used in most of the developed world. A real patronymic literally signifies that a person is the son or daughter of so-and-so. Because patronymics do not pass unchanged from parent to child, a patronymic is not a family name.
So, if your last name is Olafson, and your father’s and your grandfather’s also, then “Olafson” is your family name and it is a name originally based on a patronymic.
If your grandfather was Thor Svenson and your father Olaf Thorson and you are Knud Olafson, then Olafson is a patronymic.