Anthy from Revolutionary Girl Utena, right down to being a goddess stabbed by the villains to contain her power (remember, Rei has Lilith's soul.) Given Utena's original air date, she's probably the Ur Example, as wee as one of the few to stay true to Rei's Unbuilt Trope nature.
Lain from Serial Experiments Lain fits everything here, except she has brown hair. Notably, the creators of Lainhave said they didn't even watch Evangelion until they finished the fourth episode of Lain.
Another borderline example, Subaru from .hack//SIGN. She has short, bright blue hair, brown eyes (close enough to red), is quiet and very soft spoken as well as showing signs of being shy in RL. Also In real life, she is in a wheel chair from a terrible accident she keeps secret, and is a Meganekko. The series shared a character designer with Evangelion, so it's plausible it was a deliberate reference between the two characters.
Tsukasa shows traits of this, as well, mostly being emotionless (though it's more an angry sort of apathy).
Eureka from Eureka Seven. Very light turquoise hair, a Human Alien, very attached to her boss at first and is an Ill Girl and Emotionless Girl as a result of her upbringing, prior to a traumatic event that would trigger her Character Development, which then was accelerated much further by meeting her Love Interest. Specifically, she was used by the government as a Tyke Bomb, and trained to kill almost remorselessly, until one day, when she was checking a pile of corpses looking for wounded people to finish off, she found three orphans, had a huge epiphany, and decided then and there to adopt them and defect from the military. Realistically enough, she could still slip back into emotionless behavior on occasion. The occasions presented themselves.
Mytho from Princess Tutu is a male, magically induced example: the shards of his heart, his emotions, were shattered throughout the land. Duck spends much of the first season working to restore his emotions.
Yuki Nagato from Haruhi Suzumiya, to the point where her name is a pun on Rei's—Rei can be read as "ice", Yuki as "snow", and both Nagato and Ayanami were Imperial Japanese naval warships. Curiously, Yuki is described in the novels as having black hair (since in the Haruhi universe people have normal hair colors) but in the anime and even in the novels' art, it appears of a purplish blue similar to Rei's (but much more pale).
Merudy, a young girl with short-cut pastel-coloured hair and a petite body, is very stoic and her voice in the anime is very soft. She even calmly starts to dispatch Erza and Juvia, claiming with a cold expression that it's her mission to do so. She's also fiercely devoted to her parental figure, Ultear, going as far as to wanting kill Gray, the person she hates for killing Ultear's mother. After the timeskip, her personality has completely changed and now is very cheerful.
Yukino Aguria, a shy, obedient girl wih short silver/light blue hair.
Cheza from Wolf's Rain. She looks like Rei Ayanami with pink hair. Being half-plant qualifies her as not fully human. She's also quite emotionless, although she does perk up quite a bit in the presence of wolves.
Shimei Ryomou of Ikki Tousen (Battle Vixens). Complete with bandage over one eye.
A rare male expy in the anime adaptation of the yaoi manga Sensitive Pornograph as the Extreme Doormat character Aki. He has her hair, dark reddish brown eyes and even has his arm and right eye bandaged similarly by the end of it.
Sextum in Mahou Sensei Negima! fits most of the attributes, although she doesn't show any interest on becoming more human in her few appearances. Her twin 'brother' Fate might fit in as a male example of the trope.
Mei Misaki, from the Another horror/mystery light-novel series most certainly qualifies. An Eerie Pale-Skinned Brunette with emphasis on eerie, dark short hair, skin so pale it is in a shade of grey, red eyes, an eye-patch resembling Rei's usual bandages, a deeply stoic demeanor and the overall feeling of creepy cuteness/cute creepiness exhaled. Visually, she is basically Rei with a pitch black hair.
Kirika Yuumura, from Noir is another brunette with similarities - Initially emotionless, although she does cry in episode one (although she's crying because she can't feel regret, not because of the dozen men she just killed), soft-spoken and created as a weapon. Her eyes are at the red end of brown, getting redder whenever there's an open fire around - used to great effect when her 'true self' comes out towards the end of the series. The series ends with her choosing between being human, or being a weapon.
InoriYuzuriha from Guilty Crown shares many traits with Rei, in both personality and situation. Stoic, seemingly emotionless? Check. Red eyes? Check. Deeply devoted to her boss, and, later, the main character? Check. Artificial Human based on one of the protagonist's family members who nevertheless has boatloads of UST with him? Um...check. The only way for Inori to be a more thorough example would be if she had blue hair instead of pink and piloted one of the show's mecha.
Jo of Burst Angel. Dead ringer, aside from a rougher appearance. She's even referred to as an angel of death. And they are both Artificial Humans.
Due to their awkward habits, stoic natures, pale skin, and lack of proper people skills, L and his eventual successor Near from Death Note are rare male examples. This probably goes even moreso to Near due to his hair color.
Kanade "Angel" Tachibana from Angel Beats!. She's presumed to be an angel working under God to oversee the afterlife. She's also seemingly emotionless, to the point where the other characters weren't even sure if she was sentient at first. She's actually a human just like the rest of them. Her powers come from a computer program, and she seems blank because she'd given up on forming attachments to anyone. She displays some subtle humanity when Otonashi is getting to know her, and by the finale she's dropped the facade entirely.
C.C. also qualifies by the virtue of unusual hair and eye color, troubled past, stoicism, immortality, and gradually warming up to Lelouch up to a Heroic Sacrifice for his sake (much like Rei, she gets better). The main things that set her apart are her tendency to mess with people and her pizza addiction.
The World God Only Knows: Dokuro, the girl who Keima meets in his time travel and the past self of his teacher Yuri Nikaido, has many characteristics of this trope. She's a powerful Artificial Humanand some kind of clone of the original Dokuro Skull, has bobbed hair and Purple Eyes. When she's introduced, she's an Emotionless Girl, but thanks to the main character she warms up and becomes very upbeat.
Mikasa Ackerman from Attack on Titanplays with her status as one. She fills some qualities of a Rei Ayanami expy quite well (fairly emotionless, dark and troubled past, short cut hair, teenager, named after IJN Dreadnought), double subverts others (her hair color, appearance and skin tone are all considered unique, but that's merely because she's the only asian woman alive, which also gives her immunity to the Coordinate), and defies others (is never injured thanks to her phenominal combat skills, is the lead protagonist's adopted sister and had no idea that she was related to Kenny Ackerman).
Pacific Rim gives us Mako Mori. While being older, having a rather more well-adjusted and sociable personality and there's nothing artificial about her origin, she's the only Japanese character in the film, is stoic and reserved, fiercely loyal to her adoptive father figure, and sports Rei's haircut as said above. It's even colored blue, though only at the tips. She also happens to be a Humongous Mecha pilot. Popular fan consensus is that it's supposed to say, "We took -some- inspiration from Evangelion but didn't try to ape it".
Explicitly invoked in the Japanese dub. Instead of having Rinko Kikuchi dub her own lines over in Japanese (which she would be perfectly capable of doing), Mako is voiced by the same person responsible for Rei - Megumi Hayashibara. You can't get to be a bigger Expy than that!
Devil Survivor's Amane Kuzuryu, a blue haired, blue eyed Emotionless Girlthat serves as the host to both an angel and a demon. In her route, she reveals the reason for her behavior and wishes that she had been able to lead a more normal life.
The Character Kos Mos in the Xenosaga Series is a female blue haired, red eyed Emotionless Robot which becomes more humane/defrosting. Like Rei, she and her successor Model "T-Elos" have got something to do with the plan of the antagonist/government organisation.
A better example would be the dancer Ninian from 7. Though she lacks the short hair, it IS blue, and she has red eyes, an unusual color even for FE standards, as well as pale skin, and is pretty frail both in cutscenes and in combat. She's also notably shy and reserved but can open up to all 3 of her support options (one of which is a possible love interest), and hides a dark secret She's a dragon born from a human father. Her tragic family history is revealed in Hector's story, in which the player learns that her father is the main antagonist, Nergal, who sent her and her brother through the Dragon's Gate for their own protection during the Scouring, a massive human/dragon war. Their mother, Aenir, was killed, and Nergal, seeking to reunite with his children after the war was over, began experimenting with unscrupulous dark magic that warped his mind and made him forget his original purpose. Centuries later, in an ironic twist of fate, he manages to call his children back through the Gate, but they don't remember him, and he doesn't remember he had children in the first place, only wanting to use the siblings to further his own goals of obtaining power. Worst of all, he actually laughs when Eliwood accidentally kills her.
She also happens to predate Rei by a year. One Must Fall came out in 1994. Evangelion first aired in 1995.
Blaze the Cat of Sonic the Hedgehog fame is a sort of anthropomorphic interpretation of this type of character, down to the same general color scheme and hairdo. Because her role as the dedicated guardian of the Sol Emeralds is of tantamount importance in her dimension, she initially distrusts strong emotions and social attachments, which she sees as distractions. She later embraces both to a degree.
Stern the Destructor of the Magical Girl Lyrical Nanoha As Portable games. She's an ancient, alien, Magi Tech program that copied Nanoha's appearance, with a few changes such as her hair being cropped short and her eyes being an unnaturally light color of blue. As the Material of Wisdom, she has little to no emotions and is completely logical. However, she starts gaining some emotions after she met and lost to Nanoha, even gaining her own personal desire to improve herself in the hopes of one day beating Nanoha in a Lighthearted Rematch.
Shiki from the Samurai Shodown 64 series is older than the average, but otherwise fits the bill nicely (the whole Technicolor Eyes thing being derived from Demonic Possession, but she has heterochromia meaning she's not completely being controlled).
Courtney from Pokémon Ruby is re-imagined along those lines in the remake, Omega Ruby, with robotic mostly short speech patterns, (though she does speak full sentences at certain points, though with William Shatner levels of pauses) violet hair and eyes. Ironically, different traits, such as short hair and an attraction to a protagonist are only apparent in the Ruby and Sapphire chapters of Pokémon Special which is based on the original games and thus is not an Emotionless Girl there (or in the original games for that matter). In both cases she does have a connection to the antagonist, being a member of Team Magma and one of its Co-Dragons.
Primula of SHUFFLE!. A young Emotionless Girl with very light purple hair and pale skin. In the anime, emotions cause her magical powers to go haywire. In the visual novel they're hoping she can gain emotions to show the program she's part of is a success, putting her into a human environment is part of the program, her magic has little to do with it. In both she is an artificial lifeform.
Kyouko Kirigiri of Dangan Ronpa. Purplish white hair, purple eyes, stoic and mysterious (even her talent is just listed as "???"), she has some unresolved issues with her father and was also secretly amnesic the whole time. She also gradually warms up to the main character.
And of course, Ruri Shiromizu from Season of the Sakura.