When your first and last name could be thought of as a what is the word for that?

The word for when your first and last name could be thought of as a phrase is alliteration.

Alliteration is the repetition of the same consonant sound at the beginning of words that are close together.

For example, "Peter Piper picked a peck of pickled peppers" is an example of alliteration.

If your first and last name begin with the same consonant sound, such as "Sally Smith" or "Bob Barker", it could be considered alliteration.