Do you think it should be called bag-power-set rather than bag-power-bag?

I guess I'm inclined to think that the operation is called "powerset" even when it's generalized to multisets/bags. But, the text "The procedures for creating and manipulating bags are the same as those for sets, except that set is replaced by bag in their names" means that bag-power-bag is the more consistent name. Personally I don't have strong feelings either way and this feels a little like bike-shedding. I'm comfortable with either name.

Duplicates are treated like any other elements: the power bag of
{a, a, b} is {{}, {a}, {a}, {b}, {a, a}, {a, b}, {a, b}, {a, a, b}}.
I'm not sure what there is to clarify: can you explain?

That is what I'd expect, but a reader might be forgiven for thinking that a power bag is the same as the power set of the unique elements of the bag. In other words, it might help to clarify that the power bag is a set of bags, not a set of sets.

Best regards,
Kevin Wortman