SPOILER WARNING FOR MARVEL MCU
As voted by patrons in this season’s poll, we’re talking about Loki and platonic love in the MCU! Note that this is the cinematic universe version, so there will be inevitable contradictions to comic lore. Also, I won’t be mentioning the Loki series in this meta at great length, as I will be focusing more on his familial relationships.

Friends
It doesn’t appear that Loki has much in the way of friends on Asgard, at least outside of his family. His reputation for mischief was established very early and it soon snuffed out any chance for that, most likely. The people of Asgard were also probably drawn to Thor more than Loki as a result as well. Given how casually Volstagg and Fandral mocked Loki in the first Thor movie, Thor’s friends wouldn’t be stopped if they teased him either. Until Mobius in the Loki series, it’s all too possible he didn’t have anyone to call friend who wasn’t also family. He had a friend in Frigga and to an extent, Thor too, just by his nature. It is more subtle and unspoken in the cinematic universe, which I’ll delve more into later. First, there’s Frigga and Odin.
His Adoptive Parents
Odin was alone when he found Loki on Jotunheim as an abandoned baby, and it’s safe to assume he didn’t have a chance to speak with his wife, Frigga, before bringing this baby boy home. We can speculate as to Odin’s complex and layered reasons for bringing this infant into his home to raise as his own, but the core of it was that this was a child in need. Odin knew that and knew Frigga would agree, so there he was. Concealing his lineage to make him appear Asgardian kept him safe, and focusing his attention on Thor afterwards presumably had two purposes: readying Thor for the throne and keeping Loki safely unnoticed.

Frigga did all she could to counter this pain, becoming his favorite parent and his first experience with unconditional love where he could recognize it and not question it. Where he struggled to get Odin’s recognition and appreciation, she gave it freely and told him he could accomplish anything. Hers was a love he didn’t have to rationalize or earn, it was there for him to observe and believe in. For Loki, who was already analytical by nature, not having to be on guard with Frigga would have been a rare and tremendous relief. Not only did she believe in him from the outset, he could get the affirmation and support a child needs when he learned sorcery from her. Instead of feeling overshadowed like with Thor or unrecognized like with Odin, Loki found a safe place to be genuine with Frigga.
But one person can only do so much to compensate for the neglect of another, and Odin becomes a figure of denied love for Loki.
His Brother, Thor
With Thor, this gets trickier and more compelling for it! Loki’s feelings for him are complicated and contradictory, and the incident where Loki turned into a snake to stab him is a perfect demonstration of that. Thor himself says Loki only turned himself into a snake “because he knows how much I like snakes”, and he wouldn’t care to remember something like that about someone he doesn’t like at all. Going beyond that, if he was capable of turning himself into a snake, he was capable of other illusions too. Perhaps not being seen at all or having a copy of himself distract Thor while he stabbed him. The point is that he didn’t need this information on what Thor liked to pull off this violent prank (Asgardians, what can you do?).
Thor, through no fault of his own, became the representation of everything Loki struggled with. He was to be king, while Loki was to be nothing. He was well-liked, while Loki had no friends to speak of. But he was also kind to Loki and played with him throughout their childhood. And in his way, his affection for Loki was also unconditional! Because even after stabbing him, turning him into a frog, and a whole upbringing full of Loki’s antics, Thor always saw him as a sibling. If we lean on comic lore here to fill in the blanks, Thor always respected whatever pronouns Loki preferred, considering him a beloved sibling regardless. A fan comic I adore does a pretty good job demonstrating how this has gone in the comics, haha.

Self-Fulfilling Prophecies in Feeling Unloved
Odin was the first person Loki really developed these self-sabotaging thoughts around. In psychology, a self-fulfilling prophecy is defined as “the phenomenon whereby a person’s or a group’s expectation for the behavior of another person or group serves actually to bring about the prophesied or expected behavior” (source). This kind of behavior in Loki was fostered by Odin himself, unfortunately. He would give Loki scraps of observable affection where it was safe to do so, which naturally led to Thor appearing to be loved ore. That always left Loki wishing he had more. Then wondering if perhaps he doesn’t deserve it. Then spiteful for being made to feel that way. Then turning that spite on Thor or get up to some other mischief rather than risking what little love and positive attention he got from Odin by turning it on him. This cycle made a foundation for Loki believing he was unloved by Odin and unable to take the risk on thinking otherwise.
One Prophecy Never Fulfilled: Loki and Thor
But just as Loki paid attention to the things Thor liked, Thor also paid attention and knew Loki better than anyone else. After Frigga’s death, he conjured an illusion that concealed his despair and how he’d destroyed his cell. During the entire MCU series, Thor shows absolutely no magical ability to see through Loki’s magical illusions whatsoever. So when he calls him out as using an illusion, it is his personal knowledge of Loki and his love for their mother, as well as his hidden deep-feeling nature, that lets Thor know that was not real.
The banter from Loki starts almost immediately after his release, which is just another way their companionship as brothers is shown rather than stated. They were both grieving and while Loki’s playfulness was probably also a little of escapism from that pain on his part, it was also to coax Thor into playing along. He goaded and prodded him with transformations into Sif or Captain America, teased him with being covert instead of using brute strength, and so on. So when Thor pretends to give Loki a weapon and instead, he locks him in cuffs, his parting remark that he usually enjoys tricks is more than a taunt. It’s Thor playing along! Even with Loki frustrated and unimpressed by being shackled, it was an expression of Thor’s gratitude in its own way for Loki trying to cheer him up.