No, the camera will always follow this format. You can rename afterwards, but keep in mind you will run into problems if you use a program that requires the original directory structure.
I find it best to keep them as is and put PRIVATE folders into named directories. Then most NLEs allow you to name clips or add meta data so you know what is what. At the very least, I would always keep my backups as is, then do whatever with a working copy.