Is what your brother did—expressing his desire to marry his paternal cousin after she was engaged to another man (without a marriage contract)—and his supplicating to marry her, Islamically permissible? And is there anything forbidden or aggressive in his supplication?

It is not permissible for a Muslim to propose to a woman who has already received a proposal from another Muslim, unless the first suitor abandons his proposal or grants permission. This is based on the saying of the Prophet (peace and blessings be upon him): "(The Prophet, peace and blessings be upon him, forbade some of you from selling over the sale of others, and a man from proposing over the proposal of his brother, until the first suitor leaves her or the suitor gives him permission)." Scholars have unanimously agreed on the prohibition of this if the first suitor has been given a clear affirmative response. If your brother expressed his desire to marry after someone else's proposal, this is forbidden. He must seek Allah's forgiveness and regret what he has done. He should inform his uncle of his regret and advise him to honor the first suitor's proposal, unless there is a legitimate Shariah reason (not to do so). He should also generally pray to Allah for a righteous wife, not for a specific woman.