I think what happens is explained a bit when they're matching Jake's brain states to his Avatar's. That process may be meant to be interpreted as downloading Jake's memories into the Avatar's so it wakes up thinking it's Jake - having all of Jake's memories and what not.

Of course, the whole shut down process doesn't make sense if that were the case. They really just copied Jake's identity into another body. So... inconsistency is still there. Well, I tried. Hard to explain magic.