The cab driver

The cab driver is Charon/Death/Ferryman, there to take her into the afterlife, or at her choosing, to take her back to the land of the living, which is what she chooses to do. Notice that in either event she fails to pay him and promises to return (which she's lead to do in a round about way via looping). The Sisyphus story they reference in the film while looking at a plaque briefly summarizes the story of Sisyphus cheating Death. Join that to the mythology of the Ferryman/Death requiring a toll to ferry people across the river Styx into the afterlife, and additionally join that to the Greek mythological notion of those who die drinking from the river Lethe to forget their previous life before they're reincarnated (e.g. corresponding to Jess forgetting each loop when she's "resurrected" on the beach at the end/beginning of each loop). Even if Death doesn't recognize who she is, it's because time has reset, but my guess is that he already knew the first time she died and his role requires him to play innocent.
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I'm something new entirely. With my own set of rules. I'm Dexter. Boo.

I don't see why the cab driver would remember her. From his perspective, that's the first time he meets her. She, on the other hand, has met him before, but her memory loss appears to have made her not remember that.