This sounds cool. What else out there is like it?
Evangelion both influenced and was influenced by a massive feedback loop of pop culture spanning roughly the last six or seven decades that is honestly too intimidating to even try to get into here. For starters, though, if you're at all familiar with the concept of movies, you've probably heard of this little indie series called Transformers, which is itself based on a number of animated TV shows about giant robots engaged in perpetual civil war and the little humans who help them out. Transformers is a property of toy company Hasbro; the thing about anything mecha is that it you're pretty much guaranteed to sell A LOT of toys.)
If you just want to get your grubby little paws on some of that real, real good mecha anime shit, there are a few easy places to start. You can always go with the classics, like 1979's Mobile Suit Gundam, the very first Gundam series that launched a similar global phenomenon about battle robot suits piloted by genetically enhanced psychics who fight in space against a colonizing force in the future. Like Evangelion, Gundam merch continues to sell through the roof, and plastic Gundam figurines even have their own term! (Pro-tip: A lot of the Gundams are available on Hulu.)
Netflix recently rebooted Voltron, an American mecha series from the 1980s about a team of people who pilot a giant super-robot made up of a bunch of smaller robot vehicles that can combine themselves together. Sometimes the vehicles are animals, like the lion-robots of the first series, and sometimes they're more like the cars of Transformers. You can also try out Gunbuster, a very Ender's Game-y mecha anime about a team of girl pilots who learn how to control giant robot suits, that was Evangelion director Hideaki Anno's directorial debut. If you're looking for another modern series, Hiroyuki Imaishi (an animator and animation director who worked on Fullmetal Alchemist and one of the Rebuild movies) created the hilarious, slick Gurren Lagann in 2007: an anime series about a young boy in a dystopian future who discovers a giant robot head buried in the ground that fights with a big drill. Normal and cool! (And available on Netflix!)
Guillermo del Toro's 2013 movie Pacific Rim, about giant robots powered by teams of humans that fight off giant alien beasts from deep in the ocean, borrowed a ton of concepts from the genre, and is said by the director himself to be his own love letter to the stories he loved as a kid. I can't necessarily say that the 2004 reboot of classic sci-fi series Battlestar Galactica was influenced by any anime, least of all Neon Genesis Evangelion, but that show took a similarly daring (and controversial) approach to telling a sci-fi story steeped in recognizable religious allegory and imagery. It has a cool name, though, and that's very anime.