Odin never adopted Loki because he felt sorry for the baby abandoned on Jotunheim.
"You were an innocent child."
"No. You took me for a purpose."
Loki sees through this lie straight away. He knows Odin would never rescue a child, especially that of his enemy, without an ulterior motive.
Even the reason Odin tells Loki doesn't really make sense. If he wanted to 'unite their realms' then surely Loki would needed to know he was a Frost giant, if Odin wanted to prove that the two races could live in harmony. At the very least, Laufey should have been told Odin had his son if the plan was to use Loki as a hostage to blackmail Laufey with. Perhaps this was the plan, but Laufey didn't care enough about his son to be threatened. There are still flaws with this as well and the idea of Loki being designed to be a political pawn in plans that never took place. Once Odin knew Loki could not be used to secure an alliance, why did he continue treating him as a son?
Odin took Loki because of this gif.
As you can see, Loki is able to change his appearance as a baby. He is clearly too young to have been taught this and demonstrates Loki inate magical power. If Loki is this powerful as a baby, then he has the potential to become one of the most powerful sorcerers in the nine realms. It was this moment that we can see Odin decide that he will take Loki back to Asgard. Not as a political asset, not as a son, but as a weapon.
We can see from the vault that Odin has a habit of collecting weapons. It is not a coincidence that Loki descovers the truth about himself in the same place Odin keeps the Destroyer; Loki is just another of Odin's weapons but one he could not control or keep locked in a vault.