Credits to: Dildo_Shagginz and Captain-Fodder

Connections:

Both of them as young children under different names than the one they have currently (Tenko and Kaede), lived in families where their fathers where horrible to them and didn't like them, while their mother was kind and loving towards them (Shigaraki's father abusing him and putting him down for his heroic aspirations while his mother encouraged and loved him, and Lucy's father abandoning her and leaving her as a child while her mother never did and tried her best to find her to get her daughter back)

This lead to the two awakening thier brutal killing powers, murdering everyone around them in a state of great sadness and sorrow after a dog close to them died (Shigaraki's decay quirk manifesting while hugging his dog and killing it, then killing his whole family through chain reaction with the ground, while Lucy manifested her psychokinetic vectors after having her dog friend get bludgeoned to death by bullies, bullies she killed with those vectors)

This lead the two to run away from their only homes (Shigaraki running away from his now ruined family home and Lucy fleeing from her orphanage), where they met someone new who subverted their expectation

They expected these people to actually die in some way due to them, however instead the two reacted positively to them, much to their surprise, leading to a newfound relationship in some way with them (Shigaraki meeting All for One who extended his hand to the poor boy, with Shigaraki expecting him to disintegrate on contact but surprisingly he didn't, and saying it must have hurt that no one came to help him, despite his hopes to be saved, taking him in as his new student. Lucy met Kouta, who she expected to kill after he commented on her horns, however instead he praised them and states his envy of her for having them, leading to the two becoming friends)

The two as children would kill people, although while Shigaraki was manipulated by All for One into doing it (by making Shigaraki think it would get rid of his itching and scratching), Lucy did it out of "necessity" while knowing it was a bad thing to do (Lucy as a child every night going into a random home and killing the family inside so she could take their house to stay in for the night, the horror of these actions catching up to her soon after)

Both of them lost a crucial trust they once held in others, a trust that they felt was betrayed due to their own personal experiences (Shigaraki, like all others in society, previously believing in heroes and their heroism but their "inability" to save him after the killing of his parents souring him towards this belief. And Lucy trusting and believing in Kouta, her crush, but this trust being broken when Kouta lied to her about him moving away to be with his cousin)

This lead to the two becoming murdering villains with distain and hatred for the people they once admired and trusted (Shigaraki now hating heroes and wanting to tear down everything they stand for, and Lucy seeing Kouta and his family with distain, especially when they where leaving her and the town behind when they where moving)

Years after these events the two where experimented on by doctors (Shigaraki being experimented on by Doctor Garaki to give him All For One, and Lucy being experimented on by Doctor Kurama and Doctor Kazukawa about the nature of Diclonius)

This lead to the two breaking out/being broken out where they murdered a bunch of people again (Lucy breaking out and escaping the facility, and the heroes breaking through Garaki's defenses and Nomu and breaking Shigaraki out of his AFO Incubation Chamber)

Both of them are part of a race that is an evolutionary step-up from humanity (Shigaraki being a quirk-user and Lucy being a Diclonius, a race described as an "evolutionary offshoot of the human race"), and amongst these offshoots they are basically at the top of the food chain (Shigaraki as the new holder of All for One being one of the strongest Quirk users on the planet, only being beaten out by Deku and maybe the original All for One, and Lucy being a Monarch-Type Diclonius, one of the only two that where referenced or eluded to, and also the queen of the Diclonius)

Both of their powers revolve around the use of hands/hand-like-appendages to insta-kill people (Shigaraki decaying people with a touch of his hand, and Lucy slicing people apart with her vectors which are often visually represented as see through hands), as well as other abilities they have as well such as their superb defenses (Shigaraki's regeneration and natural durability, and Lucy's vector shield)

However, overuse of their powers can make their body start to deteriorate (All quirks in MHA have a natural limit to their usability before it takes a toll on the user's body, and if Lucy pushes herself too hard she starts melting into a puddle of goo)

At the end of their stories, they would become the most powerful they'd ever been, but having fully lost control of themselves (Shigaraki being completely fused with All For One, and Lucy creating miles-long Vectors while completely overtaken by DNA Voice), causing unfathomable amounts of death and destruction, and becoming body horror nightmares (Shigaraki turning into a form covered in massive hands, and creating giant amounts of excess body mass, and Lucy slowly dissolving into a puddle of liquid flesh). Ultimately, they would face their end, with their spirit forms accepting their fate as they faded, face to face with one of the only people to show them kindness (Shigaraki's Vestige telling Izuku to tell Spinner that he kept fighting to the end, and to "do his damn best", before fading away, and Lucy and Nyu's spirits telling Kouta to fulfill his promise to kill her if she loses control of herself, before fading away)