I would say that the Duke definitely treats his wife as an object or a possession. He does not think much of women, as he seems to think his wife should obey him in all things and focus all her attention and adoration on him as if he is the superior being. He becomes angry when she does not conform to this expectation, stating:

’twas all one! My favour at her breast,

The dropping of the daylight in the West,

The bough of cherries some officious fool

The Duke's trouble is that he sees a wife as an object that he is adding to his collection rather than as a human being who will learn, grow, have her own ideas, and interact cheerfully with other human beings. The Duke comes across as a warped human being in condemning this natural humanity and natural impulse towards human community in his young wife.

The Duke reduces his late wife to an art object by keeping her portrait for others to admire while having the living woman killed. The portrait will remain static and entirely under the Duke's control.

It is reductive to see this dramatic monologue as “Mysoginist” merely. True, the Duke seems to treat his spouses as objects, but the central point is that he treats everything as a possession—including his artwork (the statue of Neptune, the portrait of his last duchess, the mule, the gardener, the potential dowry that will accompany his next duchess, etc.  This “possessive” attitude is part of a larger personality disorder—what, post-Freud, might be called a narcissistic, egocentric attitude toward all of life.  Browning is drawing a portrait of a kind of person, with a title, to be sure, but emblematic as well of contemporaries (last half of 19th c.).  We as readers must remember that when a poem is set in a past society, it nevertheless is addressing a subject on the poet’s mind as he or she writes the poem.  If Browning were addressing only the male flaw of treating all women as inferiors, he would not have bothered with details of the duke’s disdain for art, etc. The point is that when all of Browning's work is examined, misogyny does not stand out as a major concern.