Is it permissible, in Islamic law, for a second wife to seek a divorce due to the stress and anxiety resulting from her husband's sensitive personality and its impact on her relationship with him?

Divorce should not be resorted to unless all means of reconciliation have failed. If it is possible for the couple to live together and cohabit amicably, even with overlooking some minor faults and compromising on some rights, that would be preferable to separation. If you are able to live amicably with your husband and handle matters wisely, then you should not seek a divorce. However, if you are unable to live amicably with him and fear that you will not fulfill his rights, then there is no blame upon you for seeking a divorce at that point.