Related to, but not a duplicate of, Did Tarkin know Darth Vader's previous identity?
According to this answer, Vader's identity as former Jedi Anakin Skywalker was not a publicly known fact in canon. This is borne out by most of the films—only Palpatine was around during Vader's transformation, and most other people we see finding out only know because Obi-Wan and Yoda knew. As of the time of the Tarkin novel, Tarkin has suspicions, but hasn't voiced them to anyone, and Vader himself doesn't tell anyone in the films besides Luke.
But in A New Hope, we see this exchange (emphasis added):
VADER: He is here...
TARKIN: Obi-Wan Kenobi! What makes you think so?
VADER: A tremor in the Force. The last time I felt it was in the presence of my old master.
TARKIN: Surely he must be dead by now.
VADER: Don't underestimate the power of the Force.
TARKIN: The Jedi are extinct, their fire has gone out of the universe. You, my friend, are all that's left of their religion.
Here, we see Vader openly talking about Obi-Wan being his former master! And Tarkin responds by acknowledging Vader as a former Jedi. This isn't something that Vader does with other Imperial personnel at any other time that I know of. Even in private communication with Emperor Palpatine, both refer to "Anakin" in the third person.
This seems to imply that they've discussed it before, or at least that Vader didn't think that the secret was important to keep. Is there any Legends or Canon source explaining whether Vader knew that Tarkin was aware, or why he thought it was a good idea to admit to being Obi-Wan's sole apprentice?