It's safe to say Berserk anime fans were left shocked & traumatised by its haunting finale. You get that feeling on a regular basis with Shingeki as characters are torn to shreds before your eyes. Not as extreme (or as sexual!) as Berserk at its most (in)famous, no, but you get the point. Images that won't leave will remain imprinted in your brain, after viewing--ye be warned!
In terms of their settings, Berserk and Shingeki are both set during medieval times. Aside from Shingeki's 'Spiderman' way of doing battle with gigantic threats, you're gonna get a STRONG taste of bleak realism from medieval/fantasy epics, here. Unless
Both series have many epic moments, a dark atmosphere, and a lot of blood.
Attack on Titan reminds me of Berserk with its horrifying imagery and medieval setting.
Both anime deal with Demons, revenge, battle, and are both male power fantasies. If you like one you will like the other. They both have interesting protagonists. There are three anime that are very much similar to Shingeki no Kyojin, Berserk, Claymore, and Hokuto no Ken. I would say out of the 3 Berserk is the one most similar. So I recommend it as have many others and this is a very valid recommendation.
The concept of fighting and blood being spilled in a dark fantasy world is a theme presented in both Berserk and Shingeki no Kyojin (aka Attack on Titan)
There is a moody atmosphere in both series that also involves a lot of action, drama, and emotions between the main characters. It conveys imagery in a way that is violent but fits within the fantasy realm of its medieval like settings.
The monsters in both series are also considered a dangerous threat to mankind; survival is a must.
It has that medieval feel to it as well as all despair, no happiness. Both cause a lot of suffering to the viewer.
Both series tell a story set in a world where humanity is on the losing side in a war against overwhelmingly powerful and monstrous foes and prospects for the future look as bleak as the present situation. Unless, of course, the heroes of each tale, rise to lead humanity in fighting back against these malevolent and horrifying enemies.
Both have shocking imagery, are set in medieval settings, have relatively realistic characters with realistic responses and actions, use monsters as symbols of fear, and are awesome.
Both have similar settings with an abundance of violence and gore. Death and mortality are both common themes.
They both focus on a battle between violent creatures (titans/demons). Prominent battle scenes and intense combat make both series engaging and entertaining.
The main characters are both beasts who overcome obstacles through force of will. Full of action and refreshingly neither are harems or have harem aspects, i.e. the fighting in both is gruesome and is paced well, so there is no boredom and no filler.
Dark, gritty, and violent anime set in realistic medieval-ish times with doses of dark fantasy. Includes suffering, betrayal, shocking plot twists, great action, amazing storytelling, and all time great soundtracks.
Both anime are quite horrifying and have some great characters; many of which are absolute badasses. Both have a similar supernatural role in them--whether that be demons are titans. If you like one of them, you'll most likely enjoy the other because of similar dark themes and surprises.