The Emissary of power power states that people recognize you as the arbiter of the entity that gave you the power. Could you play it though as if the entity of the power is unknown, and not easily (if at all) recognizable to others?
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My character in our current round-robin campaign is an amnesiac Emissary; even he doesn't know who (or what) he's the Emissary for. Is there an example of this in the books? No. But in the end, don't make it a matter of rules, but what you need for your story.
And this particular complication definitely drives the story along, and makes for some interesting aspects for the character:
The Marked by Power in his case shows that he has had some interaction with some great power, and it has marked him- but it doesn't give specifics. At least not easily.
All of this isn't strictly by the rules, but it makes for a good story. And though The Dresden Files is a bit crunchy, it is still very much story-based.
Every Dresden Files book introduces new exceptions and mysteries to how Harry thought the world worked. So if the GM wants to, why not? There's always some exception—as long as it's treated like an exception, and a Big Deal that makes people want to unravel the mystery.
Which may not be what the player wants. If they just want to dodge an inconvenience of the power, I'd answer "Nope! That's just how Emissaries of Power work." But otherwise, the player is handing you a golden opportunity on a platter—take every advantage to build something meaningful around it. wraith808's answer is a perfect example of this idea in action.
I've played in 2 DFRPG games where someone played an Emissary. In one the character's High Concept was "Unwitting emissary of the inscrutable Bast," but he didn't know who he was working for (he later found out and changed the HC to "Agent of the inscrutable Bast").
In the other game nobody knew who the guy was an Emissary of (well, the GM might have; there were a lot of mysteries in that game).
My opinion is that SOMEONE should know, if not the player or other players OOC, then at least the GM. But making this stuff up as you go along can be lots of fun.