As you can probably tell with the many questions I asked, I have put a lot of thought towards your case. To be fair, I think that there's a chance you could be completely unrelated to these accounts. However, ultimately, there is too much suspicion for me to unban you.
Firstly, all of the other accounts banned on your IP address are alt accounts (even the one you claim is your brother's could very well be a second account), so you could have easily created them yourself to avoid cheat detection on your main account. In other words, there's nothing indicating that they are the accounts of someone else.
On the topic of accounts, that account I last posted was one of them. It was banned about two days after yours, and I'm very skeptical of your claim that you have zero connection to it.
Furthermore, the excuse of siblings getting one banned is an excuse that I have seen for many appeals and is usually false. In this case, it can't be proven without a reasonable doubt (especially with the account that was banned ~2 days after you). I also honestly consider the excuse you provided at first ("you went to your mother's house, hopped on your brother's computer to play TF2, and then when you tried connecting to our servers, you were banned.") very convoluted and therefore possibly fictitious.
With all this in mind, I have come to the conclusion to keep you banned. I'm sorry, but you can not be cleared of this Duplicate Account detection.
Appeal declined.