The correct option is (D): Alliteration.
The figure of speech used in the expression "Matthew met Michael at the mayor's moor" is Alliteration. Alliteration refers to the repetition of the same sound or letter at the beginning of closely connected words. In this case, the repetition of the "m" sound in "Matthew," "met," "Michael," "mayor's," and "moor" creates an alliteration.