Is the interest paid by a friend considered usury, knowing that it is the same monthly interest paid to the bank, and the lender does not gain anything for himself?

Your act of borrowing with interest is haram (definition) (forbidden) and is considered one of the major sins. You must repent and return the debt without the interest if possible. However, if you are compelled to pay the interest, then pay it, and the sin is upon them.

As for lending to your friend with interest, it is not permissible, whether you pay it to yourself or to the bank, because it is consuming usury (riba). This is worse than your first situation where you were an agent for riba; in this case, you are the one consuming riba.