There are a few factors which lead to Jonas feeling alone or separated after the Ceremony of Twelve, during which he learns that he is to be the next Receiver.
First, he is given his assignment in a folder, which means that no one else hears his instructions and he cannot share this new responsibility, or even the knowledge that comes with it, with anyone.
Second, the fact that there is only one Giver at a time means that this is a journey he will be going on "separate" from his peers. This is exacerbated by the fact that he is forbidden from discussing anything about his training with others in the days to come.
Third, as the new Receiver, Jonas has to singlehandedly bear significant emotional pain, as he now becomes the only person in his community to have the memories, both good and bad, of what has happened in the past.
Once he has been bestowed with all these memories, he experiences the full spectrum of emotions—from pain to pleasure and sadness to joy—which none of the youngsters with whom he has grown up have any inkling of.
The only exception to Jonas' isolation is the relationship he builds with the former Receiver, and the two become very close.