First, it is common for a son (esp the first) to be given his father’s name as his own first name (or ‘given name’).

Also, one of the longstanding ways of more specifically identifying someone is by using the name of a parent, esp. of the father — this is called a patronymic.

So you might have someone named “David son of John”, which would also be expressed “David John’s son” or “David Johnson” (or, in English, you might also have “Johns”, “Adams”, etc).

So originally “John Johnson” would be the ‘given name’ + patronymic last name. And when you consider how popular it is to name a son after the father the “John Johnson” pattern would be quite common (just check out all the John Johnsons in a Minnesota telephone directory!). I would NOT attribute it to the mere ‘whim’ of the parents.

Of course, over the past several centuries this patronymic form became fixed — that is, the son of David Johnson would, under the old system have the last name “Davidson”. But it became the habit to hand down the older name… so that David would now give his son the last name “Johnson”.

Now there has also at times been the habit of identifying someone with TWO patronymics, usually the father’s and grandfather’s. So , if David’s grandfather was named Peter, his name with a double patronymic would be “David Johnson Peterson”. (I do not believe this precise pattern with -son is typical, but there was a time when Scottish men might have mac … mac. .. )

Typically, a person’s middle name is taken from the mother’s maiden name for the purpose of establishing genealogy.

The last name is the father’s last or “sir” name.

In poetry and prose, the use of similar sounding words is called ‘alliteration.’ That may be the term you are seeking.

When the person’s first name has reference to the person’s last name, it is simply the capricious will of the parent(s) who named the child. There is no specific term for this alliterative use in naming indiviuals.