Is saving a portion of money earned through effort and hardship, while giving a part of it to the family, considered permissible or forbidden, especially given the difference in work pressure between him and his brothers?

Mixing funds and spending from them on the family does not remove the money from the owner's possession after the expenditure. If the amounts differ and they agree on equal entitlement, this condition is void, and the surplus of each fund remains with its owner. If what the questioner pays covers his expenses and his household expenses entirely, then there is no sin upon him in saving a part of his money.