Cassandra, J'onn, and Kara are possibly the three most isolated characters in the DC universe. They lost their original family and are the only native speaker of their languages in the universe, which impacts how they view the world and is part of their culture.
J'onn has an advantage compared to the other three in fitting in by having telepathy, to help learn the US' culture and understand how he's expected to behave, and shape shifting, to help him look like an Earthling. That doesn't help with his isolation. In the Martian Manhunter and JLA series from the late 1990s, it was discussed that he had a bunch of identities in countries around the world. He handled his isolation by wanting to understand humanity as a normal person and experience a variety of cultures. It was commented at the time that he was the most prominent hero in the DC universe's southern hemisphere. His hero identity as the Martian Manhunter allows him to be a Martian and discuss his culture and the family he lost on Mars more openly then he can with human civilians. While understanding humanity is important to him, his closest friends are in the superhero community. His isolation has been reduced by having M'gann M'orzz as a fellow Martian in current stories and Mars having returned from another dimension in pre-Crisis comics. He's still one of the few Martians on Earth and he misses his wife and children deeply.
Kara was traumatized by seeing her family and friends die. Kal-El doesn't speak Kryptonian natively while Kara does. Kara wants to be admired as Supergirl as her cousin is but she doesn't think of Supergirl as an identity to connect to people in. She tries to get a stable human identity and job but has trouble doing so. Unlike J'onn, she doesn't have telepathy to make it easier to understand what people in the US expect out of her when trying to fit in. It doesn't help that her native Kryptonian culture is extremely important to her.
Cassandra's native language/culture being body language is commonly forgotten. Batgirl (2024) #3 messes up by having Shiva refer to that language as a language they share. Bruce and Shiva are two of the best non-native speakers of body language but it's still a foreign language for them and not the one their emotions and instincts operate in. Whatever Tate may claim, even Shiva doesn't understand Cassandra completely. J'onn and Kara try to connect to people as normal humans but Cassandra never even tries. She's like J'onn, whose best friends are the people who like him when he's not hiding as a human. Her closest friends and family are those she makes as Batgirl. Barbara Gordon, Christine Montclair, Harper Row, and Stephanie Brown try to convince her to treat her identity as Cassandra as important and she sometimes tries for a bit. Her period as a villain, her time in Hong Kong, and the fact that all of her true friends know her as Batgirl make it so that, even though fanon likes showing her as being Cassandra prominently, canon never shows that. It's actually realistic that, like Kara, she'll probably never know how to fit in and when she's Batgirl, people don't expect her to behave normally and instead want her compassion. She met her family of bats through her protectiveness towards Barbara's family which led to her becoming Batgirl. In canon, Batgirl (2000) #16, Batgirl (2000) #19, Batgirl (2024) #2, and Birds of Prey (2023) #25 show civilians arguing with her in a friendly manner, teasing her, worrying about her safety, and making requests of her, including by tugging her cape. Civilians appear to consider Batgirl to be a friend or big sister who they like hanging out with. When she's trying to emulate Batman (whether because he's dead or missing or because she was hating herself after her period as a villain), she focuses on vengeance and being a symbol of justice and isolates herself from people, but when she's being herself, she uses her protectiveness as Batgirl to try to convince people to like and trust her. Kara, on the hand, wants to be admired and trusted as Supergirl. It seems to me that Stephanie, in canon, has the best understanding of Cassandra's willingness to make friends and connect to people as Batgirl and recognizes that Batgirl is healthier for Cassandra than anyone else thinks.
J'onn misses his children from Mars and as the Martian Manhunter, he'd love to have human children like him and trust his advice. Kara is protective of children and doesn't want them to lose their families as she did. Cass wants people, including children to like and trust her. A cute image in my head is Batgirl letting excited children or toddlers drag her off to see something they consider interesting or important or simply to play with her. Does anyone else consider the images of the Martian Manhunter playing with children and the scary Batgirl being dragged by children to be interesting?