Symbolism, Imagery, Allegory
Yes, Saphira is a major character, and yes, she also happens to be a dragon. So what could she possibly have to symbolize? Let's take a look.
If you consider how closely linked Saphira is to Eragon (and you can't possibly get any closer than sharing a telepathic mental link—just ask Professor X), you might start to think, hmm, where is Shmoop going with this? We think that Saphira's development also represents Eragon's. For example, Saphira's birth (when she hatches out of her egg) is also a sign of Eragon's birth—not his actual birth, of course, but the birth of the person he's destined to become: a Dragon Rider.
As Saphira grows, Eragon develops right alongside her. Check out Eragon's reflections on the day of his sixteenth birthday:
At nearly six months of age, Saphira was much larger. Her wings were massive; every inch of them was needed to lift her muscular body and thick bones. The fangs that jutted out from her jaw were nearly as thick around as Eragon's fist (39.15)
Now why would we get these details on Eragon's birthday? It's clear that, as he ages and develops, so does Saphira.
One last example: the climactic battle of Farthen Dûr. Just before Eragon kills Durza the Shade once and for all, Saphira crashes through the roof and we see that "[h]er jaws were open and from them erupted a great tongue of flame, bright yellow and tinged with blue" (58.78). Is it a coincidence that, as Saphira breathes for the first time, Eragon is able to finally kill his foe?
After killing Durza, Eragon reflects that he's ready to assume the responsibilities of a Dragon Rider. Saphira can breathe fire and Eragon can step up to his destiny. We'd say they've both done some major maturing in this here book.