Does Browning imply any connection between the Duke’s art collection and his attitude toward his wife?
On the most basic level, the Duke is treating his wife and his possible new wife as art objects, collected by the Duke in the same way he collected Fra Pandolph's portrait of his last duchess, and the way he collected the bronze statue of Neptune taming a sea horse, by Claus of Innsbruck. The whole tone of the dramatic monologue reinforces the Duke's egotism and self-importance, and his belief that his heritage has entitled him to treat everyone (even the ambassador) as objects in his collection. Of course, the next duchess will be treated the same way, since he is "buying" her and her dowry, by paying the title of "duchess." Browning does a lot more than "imply" ownership; he (putatively) gave commands to have her killed. Her portrait has reduced her to a mere possession of his (the curtain is only drawn aside by him).
It is hard to improve on your own elaboration of your question. The Duke reveals his own character throughout his monologue--which is what Browning always does in his dramatic monologues. I think we can assume that the Duke is not only arrogant and egotistical but that he is also greedy. He probably has an entire palace full of works of art and wants to accumulate more and more. This is in spite of the fact that he does not appear to have a genuine appreciation either of art or of natural beauty. He values the painting of his deceased wife more than he valued his wife herself. When showing his visitor the painting and the statue of Neptune taming a sea-horse, he is careful to mention the names of both artists because the names (although fictional in this poem) would indicate the value of the works. Like a lot of other art collectors, he is more interested in the monetary value of his collection. There are many people who invest in art for the sake of future profit and at the same time to make an impression on others. The Duke here is trying to make an impression on his visitor. Evidently the Duke's interest in the Count's "fair daughter" is mainly in her dowry.
It is noteworthy that the visitor, who is sitting down, seems to have heard enough and jumps up abruptly to leave the room. This explains why the Duke says, "Nay, we'll go / Together down, sir." The word "Nay" suggests that the visitor is in a hurry to get away alone.