Robb Stark and Theon Greyjoy were thrown together due to the political circumstances that surrounded their lives when the pair were not much more than babies. They both lived and grew up at Winterfell, and, like most regular siblings, did not have a say in the fact.
However, their relationship was much more complicated than that of real siblings (or half-siblings i.e. Jon Snow) or even regular friends. They were victims of the circumstances that existed in the country in which they lived and those surrounding circumstances meant that they could never really be friends, much less family.
They Didn't Know Each Other
Theon may have grown up with Robb but the pair never really understood or knew one another. There was a lot of resentment that percolated under the surface that neither boy ever realized.
When Robb sends Theon to his home in the Iron Islands, to speak with his father Balon on Pyke, that proves that Robb didn't really know Theon at all. If he'd fully understood the Greyjoy he'd grown up with he'd have known better than to send him back to his family, even if he wanted to prove he trusted him.
Theon Betrayed Robb
If Theon actually cared about Robb, if he'd actually seen Robb as a true friend, he would not have betrayed his "brother" to his father, Balon Greyjoy.
Their relationship, having grown up together, would have been more important than the fact that Balon and Theon shared blood and a last name. Unfortunately, that was not the case, and Theon joined Balon in his quest to invade the north in Robb's absence almost immediately. That's not the kind of thing friends do.
Theon Took Winterfell
Theon, after betraying Robb to his father, Balon, goes even farther by traveling to Robb's (and Theon's) home of Winterfell and taking it for the Iron Islands. Not only does he take it but, even after being betrayed by his own men, Theon never even considers returning to Robb's service.
Of course, Theon would have complicated feelings towards Eddard Stark and his time at Winterfell, having been taken from his family, but he grew up with Robb. If they'd been friends Theon would have returned to his service instead of headings for Winterfell.
Always Arguing
In what should have been a heroic moment for Theon, Robb can't thank him or even be kind to him. When the Wildlings find Bran in the forest outside Winterfell it is Theon who rescues Bran by shooting the Wildling holding him. However, Robb is not pleased with Theon's actions.
He yells at his so-called "friend" and "brother" reprimanding him because he might have hit Bran instead. Theon and Robb could barely get along when things were going right, their friendship obviously wasn't strong.
Ward
Theon was always the ward at Winterfell, never the guest, cousin, son, or even friends. The fact that he was kidnapped, basically, by Robb's family kept the pair from ever being able to form real bonds of friendship and trust.
Theon Greyjoy was always Theon Greyjoy. He may have been treated well by Ned and Catelyn, but he was still a prisoner no matter how good he seemed to have it on Winterfell's grounds. The fact that he was a prisoner kept him and Robb from forming real bonds of friendship.
Kills Robb's "Brothers"
Game of Thrones fans know that Theon never actually killed Rickon and Bran Stark, instead, he burned the bodies of two orphan boys and claimed to have killed Bran and Rickon.
However, if he and Robb were really friends, there's no way he would have allowed Robb to even think that he had killed his brothers. That's the cruelest sort of mind game that even decent acquaintances would not have played on one another. And yet, Theon Greyjoy does so without hesitation when it comes to Robb Stark.
Kills Rodrick Cassel
Rodrick Cassel was the Master at Arms of Winterfell. He trained all the Stark sons, including Jon Snow and Theon Greyjoy, how to fight with sword and bow and arrow. He was a well known and respected teacher, supposedly. Y
et, Theon allows Rodrick to be murdered as soon as he and his men from the Iron Islands take Winterfell. He knew Rodrick and, even if he had never actually cared for him, Robb obviously did. And yet, Theon lets the man die.
Swore Fealty
It seems only right that Theon should swear fealty to Robb as the King in the North. Yet, that highlights the problematic power structure that always existed in Robb and Theon's relationship. Robb was always a Stark, the heir to Winterfell.
He was always going to be held above Theon, even before he was a king, in terms of power structures. It's very hard to form a real and honest friendship when a power dynamic like that already exists in the relationship.
Not A Stark
Even if Theon really wanted to be a Stark, in the end, Robb never would have allowed it. Despite Theon belonging far more in Winterfell than in the Iron Islands, Robb would never have let Theon do as he wished.
After Jory Cassel is murdered in King's Landing, Theon wants to go with Robb to avenge him. What does Robb do? He reminds Theon that he is not a Stark. This may be technically true, but it's very harsh at the moment. Theon and Robb couldn't support one another like friends.
Torture
Robb sends the Boltons to take back Winterfell. He knows that Ramsay has succeeded and likely knows that Theon was taken with the castle. Robb does not step in to stop the torture Theon is undergoing at the castle.
Robb's anger at Theon's betrayal is understandable, but the fact that he didn't even want to interrogate Theon, try to understand where their relationship fell apart, means there was never a real friendship there to understand in the first place.