"My Last Duchess" by Robert Browning is written in the form of a dramatic monologue, in which the Duke is conducting an emissary around his palace as part of opening negotiations with the family of the women he intends to marry after the death of his previous wife. Since the poem is written in the first person, the Duke only refers to himself as "I."
Since the narrator's wife was a Duchess, we know that he is a Duke. The single word, Ferrarra, immediately following the title of the poem suggests that he is the Duke of Ferrarra, an Italian city northeast of Bologna. The Duchy itself was created in 757 AD, and thus the reference to a "nine-hundred-years-old name" refers to the title of Duke of Ferrarra, which gave his wife the title of Duchess of Ferrarra.
The poet does not assign personal names to either the Duke or Duchess, but just refers to them by their titles.