Why are a lot of characters in anime and manga named after their mothers and not their fathers?
Expanding my comment into an answer.

In Japan, it’s uncommon, but not unheard of, for a man to take his wife’s name after marriage. It can happen for any reason, but a common reason is when a man marries into a family with no sons and his wife’s family wants to keep their name going.

In modern Japan, married couples are required to pick one partner’s family name for both of them. From a Time article:


  In contrast, Japan requires that married couples take one of the spouses’ family names, which, unsurprisingly, means that 96% of married Japanese women assume their husband’s last name.


Any children of the marriage are given that last name as well, so if the couple chooses to use the wife’s name, the children will have their mother’s family name.

In anime, of course, this is used as a plot device, and the exact reason depends on the show. For instance, in Evangelion, Gendou takes Yui’s last name, Ikari, to try and distance himself from some of the unsavory things he did in the past. So Shinji gets his mother’s name, which might have various meanings, given all the Freudian subtext in the show.