Wa alaikum salaam wa rahmatulahi wa barakatuh sister,
You are correct, we do often see advise on caring for and responsibilities towards our spouses and children, yet the same is commonly not addressed with regards to other family members, with those being of most concern being those closest to us, that is our siblings.
As I am not a scholar I can provide a fatwa in this issue so would suggest contacting a scholar or imam if you are looking for such. I can, however, provide some advice more generally on the topic.
Of course, as we know our parents, spouses and children do have primary rights over us and us over them. Information regarding such rights is readily available as you have noted. As family members, there are also rights regarding them as we are also expected to maintain family ties beyond only our parents and children. Family ties may be maintained minimally if it becomes a threat to relations somehow, or they may be more frequent and involve providing things such as financial support, especially for example in the case of an unmarried sister. Whilst they may not be obliged Islamically to provide such, it shows good character and manners to assist if one can.
Siblings should not mistreat one another or show bias as this will only lead to severed family ties which is against the principles of Islam and is therefore disliked. Beyond that it can lead to further stress and difficulties between family members beyond only the siblings compromising family ties further. In which case it is best that difficulties between siblings are resolved to avoid wider family conflict.
Essentially, it depends on the reason why a sister may feel she needs the support of her brother; whether it is a temporary matter, or something more permanent, or financial or emotional support, whether she is married or not,, the relationship with the sister-in-law..etc.. There are many factors that may influence a brothers desire to support his sister, or if he feels she has done something to wrong him, then maybe he wants to distance himself from her somewhat. However, regardless of the situation and how much support the brother gives the sister, he should not mistreat her, just as he should not mistreat any other family member or even nonfamily member. This is not acceptable as we should be kind and merciful to everyone in the character of a good Muslim.
It may be that they need to attend counseling together, either through formal measures if necessary, or even better with an imam son that the issue can be discussed with a man of knowledge who will also be able to advise on each others rights over one another.
May Allah grant you the answer s you are looking for and bring happiness and contentment in your family relations.
Views expressed by hosts/guests on this program (live dialogue, Facebook sessions, etc.) are their own and their appearance on the program does not imply an endorsement of them or any entity they represent.