Satire is a genre with the salient trait of using irony and mockery to address a topic or a behavior, among other things.
In this short story, there is no doubt that the author was using satire to criticize an overly-economical couple who was yet rich enough to allow themselves some freedoms. The target of criticism is, therefore, the upper class couple. They are being subjected to ridicule because they live in a time and place where they could (and maybe should) be using their hard-earned money to treat themselves, help their very socially uneven community, and simply reap some of the benefits that come from hard work.
The satire is found mostly in the character of Mme. Oreille, who does not work and yet is the manager of all the money in the house.
...it really pained Mme. Oreille to see any money spent; it was like tearing at her heartstrings when she had to take any of those nice crown-pieces out of her pocket; and whenever she had to spend anything, no matter how necessary it might be, she slept badly the next night.
She is so extremely stingy that she will not even let her own husband have the money he earned. She does not even let him get a good umbrella, which is hilarious. She goes as far as asking for the insurance company to cover for it- which she achieves- and there lays another cause for satire.
Therefore, a lot of people are criticized in this story with the use of satire: The husband, for not "controlling his wife" and for allowing her to manhandle him as she wishes. Plus, she holds on to his paycheck and he cannot even convince her properly to change her ways. The wife, because she is the way that she is: neurotic, obsessive over money, unnecessarily short tempered, and overall histrionic. The assurance office man is also satirized for falling under the ridiculous ways of Mme. Oreille. In all, the situations are quite odd and awkward and render themselves subject to ridicule at any given time.