As the narrator, the Duke, shows the painting of his late wife the Duchess to his guest, clearly he is very pleased with it, even calling it "a wonder". It is also clear, as the monologue progresses, that he was very displeased with the duchess herself. She took too much joy in simple things and simple people: joy equal, in fact, to her affection for him and the grand heritage he bestowed on her. He cannot take this; his ego is such that he demands her affection solely for himself, her identity to be his alone. As long as she is just as happy with a sunset, for instance, as with him and his estate, he cannot truly own her. And this is intolerable to him. He has her killed.
She lives, now, in his painting and now he can truly own her. She is a valuable piece in his collection. She is curtained, even, and only he can draw the curtain. She can smile only on him now, and she is his.
Surely the implications of this for the Duke's prospective second marriage are not lost on the guest, but just to underline the point, the Duke refers him to another piece in his collection as they move away from the Duchess. He asks him, casually as it were, to behold his sculpture of Neptune taming the sea horses. Neptune, the god of the sea, is taking ownership of those in his realm. Likewise, we understand, a wife of the Duke must be tamed and owned.
The Duke represents himself as an art collector, yet he is obviously a cold, selfish, greedy person who is insensitive to beauty. He is more interested in the value of a painting or sculpture than in its aesthetic merit. The same is true of his deceased wife. He valued her as a beautiful possession and was completely blind to her merits as a human being. He is describing his last Duchess throughout most of his monologue, but he is looking at her painting and not really thinking about the admirable young woman herself. Proof of his ignorance of real beauty and of real art is the fact that he makes a point of mentioning the names of the artists responsible for two of the works he shows his visitor. Fra Pandolf was a fictitious artist, but the reader can gather from the repetition of the artist's name that this artist was famous and consequently that his paintings had high monetary value. The same is apparently true of the fictitious Claus of Innsbruck who made the Duke a bronze reproduction of the statue of Neptune. Art collection with the Duke is synonymous with greed and selfishness. Unfortunately, this is true of many contemporary art collectors who look on art as investments.