Is it obligatory to treat the wife's sister like the husband's sister on occasions, such as giving the same amount of money as a wedding gift, if the wife insists on that, considering "my sister is like your sister"?

It is permissible for the questioner to treat his wife's sister as his own sister in the matters mentioned, and this is considered kindness and good treatment towards in-laws. However, he is not obligated by شرع (Islamic law) to pay monetary sums to his sister or to his wife's sister, and he is not required to treat them equally. So, he may pay only his sister or both of them.