Cleaning safety razor equipment

Hello. How do you clean a safety razor stand West Coast Shaving Stand 312, Chrome and a safety razor case West Coast Shaving Badger & Blade Leather Safety Razor Case, Brown? I know that you can use a toothbrush to clean the safety razor. Can you use the same toothbrush for the stand and for the case?

The stand uses some soap and water and that's it. The case can use any quality leather cleaner like Bic.