DY of course it is his mitzvah. He however genrerally doesnt have any minhagim how to light ie on the rare occasion that he may have lit he didnt wave his hands he didnt say any tefilah he didnt light based on the number of kids he may have had etc etc etc ditto for any other minhagim regarding candle lighting.
Its pashut that when she adopts practices for which her husband has no minhag she would do as her mother does.
Note that R’ Yaakov Emden doesnt bring a raayah that she followed her mother, it is pashut he just states it as fact.