As the game reveals, Haytham was originally an Assassin. He eventually switched sides. But he maintained his Assassin skills over the years, as the first few hours of the game showed.
Out of universe, it makes sense that the developers had that subtle sign on his clothing, as it would help fool perceptive gamers into thinking he was an Assassin, prior to the reveal. I don't recall there being an in-universe explanation for that though, it's possible it's just a part of the Assassin gauntlet, and Haytham didn't feel the need to customize it by removing the symbol. By the time that we the game starts, the Templars had largely killed off the local Assassin presence, so we never saw if Haytham used it to fool Assassins.