What if the real person IS Luna. And for her to survive she had made up Sol?
And that’s why Sol never will feel like a real person. Because she isn’t. And Luna struggle with that. To make a believable Sol she can hide behind.

I think overly positive characters are often harder to relate to and feel like they're real than more pessimistic, negative ones, because it's rare to meet somebody very cheery who isn't either annoyingly blissfully ignorant of other people's suffering, or... well... fake.
Of course, that can often be the key to making a compelling cheerful character. Because there must be a reason they're so cheerful in spite of how hard life is. They could be:

1. Be totally ignorant about the pain and suffering of others or naive to anyone taking advantage of them because they have grown up in isolation or outside of the everyday world (ie. very rich, grew up in a bunker, a convent or on a remote farm). In this case, they are still coming to terms with the complexities of the wider world, and their development will involve eventually having to come to terms with the fact that sometimes you have to make complicated decisions, and for some people, life is unfair. (Nia - Gurren Lagann)
2. Be totally ignorant but it's because they're just... not very bright or complex. They can't comprehend complex problems and pose simple solutions to everything. Sometimes there is wisdom and clarity in this... but sometimes it messes everything up. (Luffy - One Piece)
3. Have reached a point of such despair that they've chosen to just... ignore it. Ignore or run away from everything that makes them sad, shut down any negative conversation and smile and act happy until they feel happy and if that doesn't work, they may even flee into fantasy or take drugs! (Aang - Avatar the Last Airbender)
4. Have come to terms with the suffering in the world and chosen to focus on the positives, take pleasure in what makes them happy and help others to find what happiness they can and standing up to injustice through positive action. This is rarely a starting point for a character. This is usually where a character from one of the previous bulletpoints ends up if their character development takes them down a path that doesn't end with them as a more pessimistic, broken person. (That said, an example where this was the starting point for a character whose development lead them to being... a selfish jerk? was Phoebe - Friends)

Try thinking about extra features of her, like the way she moves or sees herself, or give her flaws. Maybe she's a cinnamon rol that is constantly hurt or has a mischievous side nobody knows. Maybe she's too calm and relaxed she falls asleep at boring places or is too sarcastic she make offensive jokes. This way you might find the key words or adjectives you need to define your character's personality.
You can get inspiration from people you know or famous people to make her more realistic.

The happy Sol persona could also be her pre-trauma character. Obviously a traumatic experience like the one you mentioned would change someone, so it would be realistic for her to take on a moodier personality.

If you did flashbacks or anything like that, you could show more of the Sol character in that way. And bits and pieces could be incorporated back into Luna as time passes/story evolves, etc

I actually think Sol and Luna are pretty good foils for each other. You just need to unpack what you've already got a little.

If you keep the multiple personality thing, you have to figure out her trigger. Why does she flip from one personality to another? It's a coping mechanism, so what does the flip help her deal with?

Also, do her personalities know each other? Or does only one know? Maybe Luna knows about Sol, but Sol doesn't know about Luna?

If you have trouble with cheerful characters, you could make it so that Sol is a little too cheerful. Like someone out of an old movie. Or you could give her some other personality tics, like OCD tendencies, or a slightly manic side. I mean, you've got her addicted to drugs. If she was really fine and as happy as she seemed, would she need drugs? Drug addiction is usually a form of escapism. What is she escaping from?

Don't know what your story's about, so I'm not sure if these ideas will work for you. Just my two cents. Hope you find a way to love your character!

I'd like to offer a bit of advice in a different vein. This is targeted directly at

"If you don't like something it means that it doesn't belong to you"

It's perfectly fine to dislike a character. You don't need to love a character for them to mean something to a story. I've plenty of characters whose guts I can't stand, but I still use them because the story would be better for them. In fact, it's better that you do have some, because if you stick with just the types of characters you enjoy, you're going to run out of variety.

As for your issue with the character, I'm working under the assumption it's the MC.... Well, I'd need more detail on how witches work in your world. If they're anything like real witches, then they need a patron of some kind. An idea could be that, when the trauma occurred, the patron gave Luna a little bit of strength to deal with it. Except, due to (reasons), Sol was born. Or it could be her mind activating latent power to create a defender.

If you want to work the "multiple personality" angle, feel free to PM me. I've some experience with DID, or Dissociative Identity Disorder, which is what the disorder is actually called these days since MPD has so much baggage attached to it. I can tell you what I have experienced concerning DID systems, alters, fictives, and some of the more common.... I hate to use this word but "archetypes" of alters that pop up due to trauma.

I highlited the personality traits for clarity.
To be fair, reading through both descriptions (and ignoring the bits related to appearance), the only thing that struck to me as the most "contrary" traits is both her names. She would be a lot more interesting character, and way more real as well, if you mixed all these traits together into one personality. Real people are often full of small contradictions that make them really hard to actually describe in a few words.

So, she can be both cheerful and cold (the opposite of cold would be warm, and the opposite of cheerful would be melancholic or plain sad), subtle and touchy, clumsy but a risk lover as well, etc etc. Think about the ways some traits that can sound slightly opposite at first glance might actually be able to coexist within the same personality if you really think about the meaning of each trait, or tweak them enough for them to be fairly similar, but with a different meaning (carefree/careless) (bold/reckless), or conditional (she's rude, but only towards strangers).

For example, you might think someone can't be both brave and cowardly, but a character might be scared to death while diving into the darkest deeps of a dungeon in order to find something they want. They're brave enough to keep going, but they're cowardly enough to start panicking each time they have face off an enemy.

You can start with a trait list, and try to keep balance between the negative and the positive aspects. For example, all of Sol's traits you stated would be set like this in that list (from my personal POV):
-Positive: Cheerful, brave.
-Negative: Cold, touchy, clumsy, subtle.
And if she's an important character, give her at least 8-10 for each category to fully flesh her out, trying to keep it in balance. Then, once you have the list, try to write it all down into a paragraph, making more emphasis in some areas than in others, and being more specific about the most important ones. That will give you a ton of ideas about her personality, story, background, and then she will start getting more real in your mind. You'll start having new ideas about her appearance and name as you do all of that.

people often choose not to be who they really are around other people. the idea of a persona is to choose an outward facing self to interact with the world.

robin williams, one of the funniest and one of the most cheerful people to ever exist dealt with depression constantly and did everything he could to ensure that no one ever felt like that around him. he was such a shining beacon of positivity every day of filming Schindler's list ended with a call to robin williams to bring the crew back to being people again.

there are a pair of twin oracles in the show america dragon long. one sees all the good things to come and the other sees all the bad thing to come. the one who sees all the good things is void of hope because knowing every positive thing that is coming ruins the spontaneity of life, and the one who only see the bad is super upbeat and chipper because she knows everything that she doesnt see will be a welcome surprise of fresh air from her dark visions

its very possible to write a great character that lives a dark life that clings to hope beyond the possibility that they could ever attain the hope they hold in thier heart