Neon Genesis Evangelion, Sidonia no Kishi Recommendations
They both deal with mysterious beings trying to destroy humanity, focus more on character development than mech battles, and are somewhat dark in nature.
Sidonia is basically EVA in space. Alien lifeforms are attacking humanity for unknown reasons and it's up to an unlikely rookie pilot to help stop them. Great and realistic action scenes mixed with a solid story and an awesome sci-fi setting.
Mysterious, powerful aliens seek to destroy humanity, who has taken shelter in a defensive base. The only way to fight these intruders is by employing humanoid gundam-like robots, requiring capable humans to pilot them. Both focus on the portrayal of human psychological conditions.
Both are human fighting unknown creatures that comes endlessly.
Both animes have people getting into massive robots and fighting off aliens that come from outer space. Both set in the future.
Sidonia No Kishi, despite its animation, are almost completely identical to that of Evangelion in every possible aspects such as, they both have similar creepy ambiance and atmosphere where, main male protagonist battles using mecha against the mysterious beings who desolated the Earth with an 'unknown' motives.
Both of the following series also have similar feels with great plot which; consequently, build up an interests among peoples.
Both animes involve mechas, aliens and genetic experiments
Both these animes are about humanity fighting in order to survive