Does the wife's brother have the right to claim the expenses for his sister's treatment, which he had undertaken, after her husband married a second wife?

If it is established that the expenses for your wife's treatment were not originally obligatory upon you, then her brother has no right to demand them from you. And according to the opinion of those who deem the husband obligated to cover medication expenses, her brother paid these costs voluntarily, not with the intention of claiming them back from you. Therefore, he has no right to claim them back from you or demand compensation for what he paid. And Allah knows best.