Ok, I just checked.
The baby was born without a soul
like Hector was, I guess. But Hectors soul, his essence (something that this baby presumably did not have), took root in that babys body immediately after it was born, so it was never anything other than Hector Hall. It was destined to become Dr. Fate.
But after the Scarabaeus, a Fate relic, was touched to the babys chest, it grew into the adult Hector Hall.
I dont think it simply aged him, I think it was Fate magick that restored Hector Hall as a living being, back as Hector Hall and not just Hectors soul in another body. The baby was a vessel for Dr. Fate and nothing more, after Hectors essence took root and the Scarabaeus was touched to it, it completely transformed into Hector Hall.
So he has his body (but aged a little), his memories, his personality and all of that just as if his body had never been destroyed. Its how Carter and others recognize him when he takes his helmet off.
Shiera Hall was reborn in Kendra Saunders. Kendra lived a life before that full of memories and experiences. When she fell into a coma and died briefly, her soul passed on and Shieras soul entered her body in its stead. When that happened Kendra lost some of her own memories (like the attempted rape of her and her mother by the cops) and picked up a few from Shiera (like how to operate the Hawks jet), as well as some of her urges and instincts (shes more brutal and has feelings for Carter) BUT she kept her own as well. Her brain, including her memories, thoughts and personality were left the same but now she has this outside influence working within her as well.
This is how Hector described his journey:
I was born without a soul, destined to be nothing more than a vessel for the real Silver Scarab, Hath-Sets agent of vengeance.
Infinity Inc defeated the Silver Scarab, destroying what was left of my body in the process.
By all rights I should have gone to my reward then.
But I didnt.
My consciousness whatever was left of me drifted through the Dreaming.
Drifted aimlessly until Kent Nelson found me.
He told me that the wheel of life that had turned my destiny was not, in fact over
but just beginning.
So what he had was something other than a soul, it was just some form of consciousness, something different. So he still has no soul, but something resembling a soul
some kind of magical Fate thing.
Oh, and the JSA were looking for the Fate-baby, there were like 4 different possible ones and one was a girl. I got that confused.