You are conflating the necessary abstraction of skills with special powers of the Exalted here I think.
All PCs in all Storyteller-System games get skills like Streetwise, or Firearms, or Melee.
In theory all of these have some reasonable constraints towards what you have actually learned.
See with Molly, our Occult 5 took some time and AP or conversation-choices spend on catching up before we could actually use it.
Similarly our Streetwise would take some time to adjust to a new city. A person with Streetwise 5 would still have a good idea how a city functions and what actors should be there, but would have to find/hear about the important places and people in the new city to apply these principles to get the full value of their Skill, regardless if they are Exalted or not.
Our Melee is a bit of an exception because we didn't actually learn it IC (Molly was propably Melee 1 or 2 from her training with Dad before Arctis Tor), so I'm just assuming our new rating comes from exceptionally strong memories/instincts remaining from the last wielders of our Exaltation, or maybe some affinity for combat itself due to having been a Slayer's Exaltation, or a Dawn beneath that.