I have had no experience with stones in dogs, but I have passed several kidney stones myself. OUCH!
The reason they like to analyze them to see what kind they are is so that they can maybe determine the cause. For example, my brother has kidney stones, and his are caused by calcium. So, he can help prevent or control them by his diet. In my case, they are oxalate stones which form as one of the wonderful side effects of Crohn's Disease.
I wish Bailey all the best, I will keep him in my prayers!!!