Is it unlawful to use my brother's money without his knowledge to pay for social insurance, and does he become a partner in it? What should I do to rectify this situation, especially since I have already bought a travel bag with this money?

Your brother's unawareness of your taking money from your mother does not affect the permissibility of that money for you. If it was a loan, then it is a debt upon you. As for your participation in insurance, if it is cooperative and based on mutual support, then it is permissible. However, if it is commercial, then it is forbidden, and it is not permissible to benefit from it by more than what you have paid.