B can infer or identify which housework it is because a typical household involves housework. So, B would readily interpret/identify the housework as the housework associated with this particular living situation or household. All living situations/households involve such things as chores, housework, routine maintenance, etc., all of which can be referred to using a definite noun phrase (using the). For example, in
I often wash the dishes with my brother
the dishes is naturally going to refer to the dishes associated with A's household.
I often mow the lawn with my brother
is going to refer to the lawn of A's household, unless otherwise specified.
I often do the wash with my brother.
Here, wash refers to wash (dirty clothes, linen, etc) associated with this house, family or living situation. If A wanted to be explicit he could say my wash. But, still, this might refer not just to A's own clothes, linen, etc, but to the wash that A is responsible for doing. If A wanted to be more specific, he could say my own wash or the wash I'm responsible for doing.
In addition, if A had says
I often walk the dogs with my brother
B can infer that A means the dogs that live with this particular family or pair of brothers or whatever. If A wants to specify only his 'personal dogs' and not the dogs of the household or family, he could say my dogs.