Your professor was teaching Industrial chemistry. At industrial (undiluted) strengths, there aren’t many chemicals that can’t damage tissue or potentially cause cancer. Constantly breathing the undiluted fumes or other exposures will certainly carry some risk in an Industrial application.
Washing clothes in a dilute peroxide solution is not going to cause cancer, therefore simply walking outside to hang your clothes carries substantially more cancer risk than the use of Hydrogen Peroxide.
Saying it causes cancer in “small amounts” is a bit like shouting at someone that stepping on a twig is destroying the entire forest…while standing next to an inferno.
Do you wear gloves when you handle your H2O2 cleaning laundry solution?
I dont, but I dont care.
I’m neither ingesting, inhaling, nor bathing in it, so I don’t care either, nor would I be concerned to wash my hands in it were it needed. Just drinking water or being outside is more than enough exposure to cancer to be worried about.