I remember one of the creators of Sonic Unleashed, when asked if SU would be Adventure 3 early in development, saying 'I thought that was Sonic Heroes'.
Whether you're mistaken as to who said it and/or where or not, it's a valid point nonetheless.
Sonic '06 was unquestionably "Sonic Adventure 3" the game, or rather it was the game that took the system of multiple playable paths each with a central character and each with a different style of gameplay that was established by the first two Sonic Adventures and reestablished it.
"Sonic Adventure 3" the story, or more specifically, Shadow's story that was begun in SA2, is split evenly between Sonic Heroes, Shadow the Hedgehog and Sonic '06.
Sonic Heroes established that Shadow - or rather,
a Shadow - somehow survived the first battle of the ARK and was still alive on Earth and in stasis in one of Eggman's labs, to eventually be reawakened when Rogue broke in and accidentally activated Omega.
Shadow the Hedgehog firmly established and detailed Shadow's origins (created by Professor Gerald Robotnik with a blood transfusion from Black Doom and secretly given the capabilities and purpose of defending the planet from the Black Arms upon their inevitable return) and back story up to where Maria cast him from the ARK in the escape pod, which is the point at which his history as seen in SA2 began. The more detailed look at G.U.N. and an expanded role for that organization, Shadow returning to the debris left over from the first battle of the ARK in Earth's orbit and the Chaotix actually hacking the computers aboard the ARK (fortunately they were able to find the computer room) and encountering Professor Gerald's message and the reveal of the Eclipse Cannon's true purpose are all references to events and elements established in
Sonic Adventure 2, making
StH the direct continuation of Shadow's story as begun in that game. The final boss fight with Devil Doom includes a buried confession from Eggman that Shadow was in fact the original, found by Eggman after the first battle of the ARK, placed in stasis and cloned, which is where Rouge found Shadow at the outset of
Heroes.
Sonic the Hedgehog established Shadow in the present after all of that hullabaloo, gradually adjusting to his place in the world and running the occasional mission for G.U.N. Mephiles attempts to fool him into a new vendetta against the people of Earth by showing him a dark future where the humanity has betrayed him and sent Omega to capture him, but by the end of the game that future is gone and Shadow is free to live his life. Execution of the game itself aside, the game established all that and depicted Shadow as slowly accepting Rouge, Omega, Sonic and co. as friends and allies, rather than cautious allies or enemies.