Account ownership

[Tutunkommon]

If I understand you correctly, there are two accounts and both of them are in your fathers name.
The terms of service allow for your father to have exclusive use of the accounts.
At his option, he can allow a minor child to have exclusive access instead of him.

He has basically gone with one account per child, yourself and your brother.
That means he is not allowed to access the accounts himself, the same as you're not supposed to use your brothers account or him to use yours.

<snip>

Not entirely true. Quoting from the Terms of Use:

You agree to these Terms of Use on behalf of yourself and, at your discretion, for one (1) minor child for whom you are a parent or guardian and whom you have authorized to use the account you create on the Service.

The and means that I can allow 1 minor child to access the account as well as myself. Since I have 5 accounts, and 4 kids, each is allowed to share one of the accounts with me, and still be legit with Blizz. Each of the children would still only be allowed to use one of the accounts tho. So if your brother is on the second account, you are not allowed to use it to 'box, according to the ToU.