The Doctor, and the Time Lords in general, are explicitly more than we can hope to aspire to. The language of the show itself describes the Doctor the way humanity generally speaks of gods, even though we get to see the Doctor at his weakest and most recognizable moments.
The Companions, on the other hand, are the people who walk beside the Doctor. Sometimes they are other Time Lords, but most often they are humans. Companion is the role we aspire to as viewers; we want to be swept up in something wonderful and beyond our control. We want the Doctor to show us the universe.
The companion path begins when the curtain is pulled back. It may well begin before the Doctor gets on the scene, but the Doctor is the doorway to understanding what you’re seeing. Formally accepting the path means passing through the threshold of the TARDIS.
You’re a time traveler now, Amy. Changes the way you see the universe… forever.
Being a companion isn’t easy. It means starting from scratch when it comes to your assumptions about how things work. It’s dangerous. But it changes your point of view on the universe and your place in it.
You learn to trust the Doctor, and then you learn that you can’t trust the Doctor. Because his priorities are not necessarily going to be yours, because he makes decisions for the wrong reasons, or just because it’s time for you to re-enter the flow of time… Whatever the reason, all companions eventually leave, but Doctor willing, they leave breathing and stronger than they were before.
That’s the secret, I suppose. The Doctor teaches you not to need the Doctor.