Bleach, when people refer to the general product you can buy in the grocery store called "bleach", is sodium hypochlorite, not hydrogen peroxide.
You can call hydrogen peroxide bleach, or a bleaching agent, but if you ask your significant other for "bleach" you're not going to get hydrogen peroxide.
Speak for yourself, we have/had a bottle of it lying around. Used for bleaching hair and as a cleaning agent. It's not Clorox, but actually says hydrogen peroxide (low dosage though).
You can call hydrogen peroxide bleach, or a bleaching agent, but if you ask your significant other for "bleach" you're not going to get hydrogen peroxide.