Considering people won't automatically know even your character's name, I don't think there's a way for them to know for certain. However, since most players will choose to leave their possessions to an heir and then reincarnate as that heir, they will often be able to work out with reasonable certainty who you were in a past life. If you deliberately obscure your reincarnation by choosing not to reincarnate as your heir (or if an enemy has killed all your heirs), and instead start in a different family or as a ward of the state, you will likely be able to obscure your origins especially if you choose a somewhat different playstyle, and don't immediately reconnect with your former community.