I think it depends on the way in which things are done, really, but I'm inclined to think YABU to think she's being unfairly treated, though YANBU to feel sorry that DN is struggling with her mental health.
Your ILs are putting a roof over her head, food on the table and paying for her medical care. Private psychiatric treatment will not be cheap, and also be on-going pretty much indefinitely. All she is expected to pay for is clothing and personal expenses, so it's not like she's having to skimp and scrape to pay the bills and keep a roof over her head while her brother is living in the lap of luxury all-expenses paid by his parents.
Asking her to do chores for GP and her DPs is not a lot to ask in return, and they may be trying to keep her active in some way. They may well have been advised to try to ensure she has some routine/things to do as part of her treatment. If her Parents are being unkind to her, making her feel like a drain on their resources or a failure/disappointment etc, then they would be being unreasonable.
As to giving her brother lots of support financially- well this is unlikely to be indefinitely, so perhaps they see this as trying to be fair given their financial commitment to his sister is likely to be long-term? As for the trip to Australia, again not necessary, but if they can afford it maybe they are just trying to give him a treat. Not necessary, perhaps a bit of an indulgence, but hardly a crime in my view.
Also many parents support their children through university and beyond if they can't get a job- the amount they give probably relates to their income (i.e. it seems a lot, but may seem entirely reasonable amount to them in relation to disposable income). They are financially supporting their DD for the longer term too. I suppose you are right, in that this may be counterproductive as it does not incentivise him to get a job immediately, but that's more of a problem in terms of his parents not behaving sensibly towards their son- I don't see how this is to the detriment of their DD though?
The other thing that comes to mind- and I say this as the younger sibling of someone with a mental illness who required a lot of support from our DPs- is that maybe they feel they spend so much time and emotional energy supporting their DD, they feel they have not been able to help their DS as much, and are trying to make up for it monetarily in an effort to "equalise" things a bit. Perhaps not required, but I could see why it might happen.
In any case, asking her to sign up for benefits to which she is entitled seems entirely sensible and reasonable to me. Why do you have problem with this?