I researched this question online and came up with a couple of answers that I feel could be relevant to your question. The seven rooms might signify the 7 stages that a person goes through in their life: birth, toddler, childhood, teen years, middle years, senior years and death.
In the story the placement of the rooms is also significant to this theory. The rooms go from east to west which represents the way the sun rises, or is "born" each day and then sets or "dies" each day.
It has also been said that the seven rooms represent the seven deadly sins: sloth, lust, gluttony, avarice, pride, anger and covetousness. The Prince definitely displays all of these sins throughout the course of the evening.
The colors also signify the following Blue – Birth, Purple - Royalty, Power, Green - Growth and Life, Orange - Destruction or Fire, White – Purity, Violet - Knowledge and Memory, and Black - Death.
Another explanation I ran across to explain the colors are:, Blue means heaven and truth, Purple means sorrow and suffering, Green means nature and hope for eternal life, Orange means strength and endurance, White means light, Violet means dark blood and Black means death or grief.
These are a few of the explanations I ran across in my research. I hope they help. As you can see there are many interpretations.