Bleach breaks down to chlorine, which in turn reacts with ammonia to form chloramine. Mustard gas has sulfur and cannot be synthesized this way.

Chlorine based bleach is just one of the things that has the label "Bleach" in your household cleaning aisle.

https://en.wikipedia.org/wiki/Bleach#Classes_of_bleaches

The only sulfur containing bleaches in that list are potassium persulfate and sodium hydrosulfite. Neither contains chlorine.

Furthermore, the synthesis of mustard gas is non-trivial. Those two chemicals are not enough to put the sulfur on a carbon chain.

If you mix bleach with just about any other cleaning chemical, you get chlorine gas, or chloroform, or some other nasty chlorine-based poisonous fumes.

(though not mustard gas)