The duke is a damaged individual who is obsessed with control. His "nine-hundred-years-old name" gives him a great deal of power and control, so he values it. To be a duke is to be at the top of the aristocratic world, with both prestige and wealth at one's command. The duke is well aware of this and expects others to be, too—it is offensive to him if people don't value what he values. He has grown accustomed to people bowing and scraping because of his status. It is what gives him his value. Not to be awed and adoring of his title is the same as personal rejection.
This points us to the second reason his title is so important to this damaged man. His inability to relate to other people except from a position of power is severely impaired. He seems to have very little interiority himself and seems to be incapable of evaluating people on their characters. Everything to him is external, such as his title or his artwork, which are displays of his magnificence. He can't, as his late wife can, value a servant or a painter for their intrinsic character traits, such as kindness or thoughtfulness. All he can "see" and evaluate are outward status markers.
Sadly, he is incapable of even discussing with his wife what his problem is, which is that he is so terribly insecure that he needs her undivided attention and adoration. He says:
Even had you skill
In speech—which I have not—to make your will
Quite clear to such an one, and say, "Just this
Or that in you disgusts me; here you miss,
Or there exceed the mark"—and if she let
Herself be lessoned so, nor plainly set
Her wits to yours, forsooth, and made excuse—
E’en then would be some stooping; and I choose
Never to stoop.
In other words, the duke has no idea even how to begin to talk to a person as a fellow human. He is so afraid to look vulnerable that he calls it "stooping" or lowering himself to speak to his wife about a problem. It is much less stressful to his ego and damaged psyche simply to have her killed and start over. He can only allude to the pain he felt to a "lesser" person, a courtier, and only after he has regained "control" over his wife.
The duke puts unusual weight on his title because he has no way to see into a person's soul, even his own—to him, he is only his external status, and that is how he judges others.