Is it permissible for a male cousin to marry his female cousin if the female cousin's mother breastfed his sister, who is the male cousin's mother?

What we understand from the question is that your mother was nursed by one of her sisters, and this cousin of yours wants to marry you. If that is the case:

If the suckling occurred within the first two years of the infant's life and consisted of five full feedings, then this cousin of yours is a mahram (definition) to you (he is your maternal uncle through suckling), and it is not permissible for you to marry him.

However, if the suckling occurred after the two years or did not meet the specified number of feedings, then it does not establish mahramiyyah, and it is preferable to avoid marriage in this case.