There are many social customs depicted in Robert Browning's poem "My Last Duchess--Ferrara."
First, the fact that her image has been painted and is hanging upon the wall reflects the nobility of the duchess. One could assume that the fact that she was able to have her portrait painted, especially in remembrance, certainly shows her importance in social standing.
The description which Browning provides regarding the consideration which went into the painting is important as well. The fact that "Paint / Must never hope to reproduce the faint / Half-flush that dies along her throat" shows how her beauty simply could not be precisely reproduced.
Another point to consider is the duchess "liked whate'er / She looked on, and her looks went everywhere." Notably, the duchess lived a very comfortable life--away from depravity, away from famine, away from all negative aspects of life.
Here, one can see that the duchess lived a customary life typical of royalty. She was regarded by all, so it seems, as above the norm.
In the end, the dialogue provided in the poem speaks to the formality by which one would expect a duchess to speak and to be spoken to. The social custom of proper treatment of a lady, let alone a duchess, is exampled.