Yeah I think there are maybe several people who know... Tarkin appears quite familiar with Vader as well, and they spent time together in The Clone Wars, and seem almost friendly by the time of ANH. It's a weird dynamic in ANH where Tarkin almost seems to hold rank over Vader in some capacity, and some officers seem to feel they're on his level or above him (Motti). Then by ESB, people are just generally afraid of him, haha.
Tarkin also discusses Kenobi with Vader seemingly being well aware of the entire story there, aside from him having kids of course.
In a recent video the Jedi and people's general unfamiliarity with them is explained away as the lack of numbers for them, so most people never met one. Tarkin emphasizes that in the last story arc of Rebs saying he knew the Jedi, not from stories, but in person. He was there to see them die too, so he knows they're not all-powerful and omnipotent like lil Annie felt they were even in TPM.
I figure not many knew, but some important people still left in the galaxy did, ultimately. I wonder if his identity never really mattered too... Like, why hide who he was? What's it matter? It's good for the story, of course, but was there any logical reason to keep who he was from the public anyway? Even if you propaganda spin the Jedi as trying to overthrow the Republic and stuff, you can always play up Ani/Vader as a hero in stopping it from happening. The Jedi instill anything from fear, to respect, to people being star-struck, to people thinking they're full of **** scammers. It'd be easy to manipulate that against them, and not need to hide who Vader is for any real reason at all.