We can identify the dragon to be Drogon because this time we get a physical description. I presume it to be the same dragon as the first dream because it is described as “the dragon,” not “a dragon.” Drogon’s egg is mentioned in the passage, as well, to make sure that we connect him to the dream. Recall our prior discussion of physical contact amplifying the bond. Dany feels heat coming off his egg when she touches it, and for an instant, she feels like she’s dreaming again. To me, this is clear evidence for a telepathic bond to Drogon at this moment. Recall again, that the magic is amplified by physical touch.
This is a transformative dream for Dany, so much so that the handmaidens immediately noticed. This is significant; the dragon symbolically heals her mind and body, burning away her fear and pain, and making her “new and fierce.” Whether you believe that it was just a psychological healing, a physiological healing by fire magic, or some combination of these with the normal healing and development of callous from blisters due to the passage of time, we know for certain that after the dream she feels better, so she was at least healed emotionally by the dream, if only partially.