Long before there was a Elena and Salvatore brothers love triangle, there was a Katherine and the Salvatore brothers love triangle in The Vampire Diaries. Stefan and Damon fell for Katherine, so much so that they became vampires because of her. She was the love that didn't go away, as years later, both brothers think of Katherine. The reason why Elena first stood out to the brothers is because she looks like Katherine.
Apart from being their first love, Katherine represents their first betrayal and heart-break. Katherine isn't a heroine, she's often the foe who manipulates others for pure enjoyment or to get her way. For years, Katherine focused on survival, and she would do anything in order to ensure that she survived. However, she became fascinated by both brothers. While she had good chemistry with both, which one was a better partner for her?
Damon: Things in Common/The Need for Survival
Katherine and Damon have done horrible things in order to survive. In the first season, Damon actually kills Lexi, Stefan's best friend, so that he can pin all of his murders on her. This is a way for him to survive and be in good with the community. Once, he also left Enzo, his good friend, to be tortured for ages while he escaped.
Survival has always been paramount for Katherine. She has sacrificed others for her survival, just like Damon. In addition to that, both have embraced being considered the villain. They don't really care what others think of them.
Stefan: Doppelganger Connection
While we mainly focus on Katherine and Elena being doppelgangers, we find out that Stefan is also one. Being doppelgangers of the lovers Silas and Amara, they are automatically drawn to each other. Katherine even recalls the first time that she met Stefan and how she felt pulled to him like she had to none other.
This pull causes doppelgangers to be linked together, potentially echoing the love that Silas and Amara once had.
Damon: Wanted to Spend Eternity with Her
Damon chose to be turned in a vampire in order to spend eternity with Katherine. He was fully aware of what she was and accepted her. Despite the difficulties that would face him as a vampire, he didn't care as long as he was with her.
When he became a vampire and Katherine was gone, he didn't want to complete his transformation. At his core, Damon, is a romantic, and he once fiercely loved Katherine.
Stefan: Stayed Attracted to Katherine
Despite how many times Stefan tells everyone that Katherine forced their connection, compelling him to fall in love with her, he remains attracted to Katherine over the years. In the first season, we see that Stefan still has a portrait of Katherine in his room.
Then later in Season 5 when Katherine becomes human, they become intimate once more. As much as Stefan seems to regret his past liaisons with Katherine, he keeps on making them.
Damon: Owned His Feelings
Whereas Stefan says that whatever he felt for Katherine was false and that his love was never true, Damon owns his feelings. He admits that he fell in love with her, and it wasn't due to being compelled.
While his love does seem to dry up quickly once she is back, he loved her for hundreds of years. Despite romances with others, he waited for Katherine. He didn't pretend that his feelings were false.
Stefan: Share an Understanding/She Knows Him
Yes, Katherine is a master manipulator, but the reason she is because she can read people well. In Season 5, Stefan tries to come to terms with being locked away for a summer by Silas while his brother and former girl-friend had the delightful summer of their lives. They didn't think of him, and they became closer. In Season 5, Stefan has a lot of anxiety, and it seems to be about being confined in tight spaces; however, Katherine knows that this isn't true.
Out of all of his companions, she is the one who calls him out and speaks of the buried truth. She tells him that he isn't dealing with his pain and trauma. The real issue is that Elena chose Damon over him. Once again, she is right, proving that she understands Stefan more than most do.
Damon: Would Have Done Anything for Her
In order to get Katherine back (when he thought she was locked away), Damon would have done anything to get her back. He practically did do everything for her. She is the main reason that he is back in Mystic Falls; he wants to save her.
Although he is enamored with Elena, his main focus is still on Katherine. . .until she comes back, of course. She isn't what he imagined, but had she told him that she loved him, he would have been her puppy with razor-sharp teeth.
Stefan: Was Kind to Her When She Was Human
Turned human by Elena forcing the cure on her, Katherine isn't sure what to do. She actually tries to die by suicide. Stefan finds a note that she left her daughter, and he goes to save her--catching her when she jumps off the clock tower before she hits the ground.
When she tells him that she is aging, he tells her the thing that she needs to hear: "You're Katherine Pierce. Suck it up." Katherine is strong, and she needed to be reminded of that strength. Stefan displays kindness in that moment, and in many more in Season 5.
Damon: Waited Hundreds of Years for Her
Damon was extremely loyal, always looking for a way to save Katherine. He believes that she is one of the tomb vampires desiccating under the church. Damon has believed this since he was turned into a vampire. All of his vampire life has been focused on how to free Katherine.
Later, he finds out that she has been free all along and just never looked for him. This would sting more if Damon found out that Katherine kept checking on Stefan over the years to make sure that Stefan was okay. She didn't do the same for Damon.
Stefan: Sympathized with Her Painful Memory
When Katherine was first dying in Season 5, she kept on remembering a very painful memory. When she had outsmarted Klaus, he decided to take it out on her family. He brutally killed her mother and father. Young Katherine, newly turned into a vampire, comes back to see blood smeared about the room, and her loved ones dead. She keeps on returning to this memory. Damon enters her mind and tells her that everything that happened to her family was her fault and that she deserved everything bad that happened to her.
Stefan enters her mind, tells her that it wasn't her fault, that she was just a 17 year old girl, and didn't deserve what happened to her. He took away the blood and the bodies in the memory, and added in the cry of the newborn she wasn't allowed to raise. While Stefan was the one who ultimately killed her after she came back to life again, he also was the one who showed her great kindness. This alone could prove that he was the best one for her.