If you take the time to thoroughly rinse your brush after use you won't have to clean with any soap. It's only when you don't and the soap and residue starts to build up that you have issues. The only caveat to that is if you have water quality issues and then the minerals and soap can build up and it's almost impossible to rinse out and then at some point you need to give it a good shampooing.

A brush is after-all hair so pretty much anything you use on your hair you can use on the brush no matter if it's badger or boar or horse.