Why am I not eligible for a refund?

There is no way to prove it was your brother or you who purchased it. If you were to get a refund that would open the door for millions of "it was someone else" claims.

You could try calling or chatting with someone from xbox support just to check, its always worth talking to a person. If you have had a refund recently it could affect this one though. As well as the fact that xbox states all sales are final, so returns
are not a sure thing. While it may meet the criteria for a return I do know you cant just return a game simply because you do not like it. That being said if you selected that you did not purchase the game in the return request I would highly suggest calling
and seeing what they can do.

Just reply to the answer that was send to you for more info. To quote from the support page: If you have a question once your request is processed, reply to the decision email"

Dear Jimo1996,

Many companies that sell games, apps and programs tend to have a time period and once the time period has exceeded then they do not refund.

Have you tried contacting the site from where you purchased to ask indepth of w hy they are refusing to refund you. Because if it is due to time period elapsing then it is as I mentioned. However they may give another reason which you could then discuss
with them to see if they are willing to understand.

Please let me know what happens

Have you received other refunds this year? What region are you in? When you filed for the refund what reason did you give?