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
Broke in the orchard for her . . .
as if she ranked
My gift of a nine-hundred-years-old name
With anybody’s gift.
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.