Salam yahya
Because the issue of disagreements between men has been adrressed in the previous verses in the same sura when Allah was saying that if the Prophet were to listen to all of the believers' whims, then the belivers would have been in trouble but Allah has made the love of faith stronger and dearer to the true believers than their ego and their need to win arguments. Allah has asked the Prophet to take just decisions and side with the beleivers who are more just in a disagreement between two parties of believers.
In the surah it is being alluded that just because they are believers, no one will be favored, the Prophet is asked to fight the oppressor party among the two till justice is served. When the issue of disagreements between two believers arise, Allah is letting us know He wants the Prophet to be on the side of the righteous person.
We often forget that the Prophet was often asked to mediate between two or multiple feuding parties. To define the role of the Prophet during such conflict between believers and also to inform/assure the believers that the Prophet has been instructed to be fair and just no matter who is the person feuding. This is very important because maybe some people's hearts thought that Prophet will favor those believers who were personally closer to him or more socially powerful and rich believers will be favored in conflicts. But God leaves no scope for such a doubt in belivers' minds when He says that being a believer does not automatically grant u Allah's and the Prophet's favor, if u r justified and u r the one who is correct and just, the Prophet will side with u.
Then Allah moves on to say what kind of a relationship He expects believers to have. He expects believers to behave like brothers ( merciful, forgiving and kind) to each other despite having differences among themselves. God then clarifies that women as well should be just and not oppress, during those times most women were not involved in the structuring of a society i.e the judiciary etc. Women were in charge of household matters but everyone knows that petty politics and fights also occur between women in those surroundings, some women domestically abuse other women, they call each other names etc. So God was asking all people be they men or women to be just and not oppress each other.
The following verses then say that if believers continue to backbite and slander and be mean to each other, why do they do such things? Would they like to eat the flesh of their brothers? Why is brothers used here? Does it mean people only backbite about men? People don't backbite about females? Obviously the answer is no. Males, females, all people backbite! But the point is to Allah it is like we are ripping into the flesh of someone who is dear to us. When we read that verse we can understand just how abhorrent God finds such behaviour among believers.
Then Allah continues to address believers and tells us that all people come from the same initial pair of humans, how we all have the same roots and our purpose is to get to know each other and be kind and just with each other, God divided us amin different nations but no nation or people or tribe is superior or is dearer to Allah. How the most noble/righteous among us are the dearest to Allah.
To summarise, mentioning women seperately does not mean anything sinister exists among women or something like that. It just means Allah is clarifying that He expects women to not be bullies to each other and be just. Generally women dealt with the household issues and domestic issues, mainly Allah is saying no matter in which sphere of life ( social life, professional, domestic household, judiciary etc) there is a disagreement between believers, the believers are expected to be just even if it means going against family members and friends. If the Prophet is asked to be the judge, then he is asked to not side with the oppressor.
The verse u cite is not singling out women with any purpose in mind. We always have to read surrounding verses to get the hang of the context. Believing women fight among themsleves, believing men also fight among themselves and so do believing men and women. The verse is not singling out women as more aggressive or as more prone to fighting or as more unjust or cruel among the two genders. I believe that the intention was to bring women and men under the injuction of justice, forgiveness and mercy.