Short answer : Miss Emily's relationship with Homer Barron is not at all like relationships in the South during her time.
Whenever one discusses a work of literature, consideration must always be given to the setting and the action that takes place within the frame of this setting. Therefore, in a discussion of Miss Emily and Homer Barron's relationship, any critique of it must be made within the conventions of the time period of Faulkner's story. That the old men who attend the funeral for Emily dress in their Confederate uniforms helps to establish the time frame: approximately 40-50 years after the Civil War. Also, there is mention in Part I that the taxes for the family were remitted in 1894, when Emily was a girl.
At the time of Emily's youth, then, there was yet in place a class system. She is part of the upper class in the South, where were few would be considered middle class as the South did not become industrialized until the early 1900s. Birmingham, Alabama, [named after Birmingham, England, then the major supplier of iron goods for its country] lead this industrialization because it had everything to make steel: iron ore, coal, limestone, and dolomite. Essentially, though, the South remained an agrarian culture with a strong division in classes as many of the industrial workers, such as the miners in Birmingham, were little more than poor whites or inmates leased for the work by prisons. They could have also been Italian immigrants from the North who came down to Birmingham, who would certainly have no social standing.
So, when Miss Emily Grierson, from the aristocracy of the Old South, begins to be seen with Homer Barron, "a Yankee" as they would call him, the townspeople are appalled. For, they are still in pain over the deaths of loved ones (every single family in the South lost at least one male family member in the Civil War) and the devastation to their land in the War Between the States; moreover, they are yet very class conscious. This sense of social class is the reason that the minister's wife writes to Miss Emily's relatives in Alabama, and the townspeople are "really glad because the two cousins were even more Grierson than Miss Emily had even been." Further, the narrators remark, "So we were not surprised when Homer Barron...was gone."