Is it permissible to take from the amount that the colleague relinquished to my brother, in exchange for previous debts of the brother that the questioner paid from his own money, knowing that the colleague relinquished the amount thinking it was a rent debt, while it is in fact a personal debt?

If your colleague gifted the remaining amount to your brother, the gift is valid. It is permissible for you to take the remaining amount in exchange for your brother's previous debts that you paid on his behalf. This is considered a legitimate set-off (muqassah) if the debt he owes you is equal to or greater than this amount.