To me, it comes down to what part of an object we view as its identity.
Theseus's ship is a ship. That's the form the object itself takes. And there are many ships. And we could make more just like it. None of them, however, would be Theseus's ship save the first. The ownership is part of the identity of the ship. But not all of it, as another ship owned by Theseus wouldn't still be considered equivalent to this ship.
How about the wood itself? Well, doubtlessly the ship underwent some repairs during its time, so much of the original material is gone regardless of any further restoration. Yet we still consider it Theseus's ship. So the material is important in that it still looks and feels like Theseus's ship, but the individual identity of each component isn't necessary to the holistic identity of the ship.
And, to me, the answer is what I call the "Darrin Principle," after Darrin Stephens from Bewitched.
Darrin's character was played by two different actors. Dick York played Darrin first. A back injury made it impossible for him to continue, so he was replaced by Dick Sargent. For the purposes of the show, however, Darrin Stephens was still Darrin Stephens. The other characters behaved as though he was Darrin, and he still had the same personality and role in the show. The character was being played by a different actor, but one similar enough (even in first name) that the concept of the character was able to continue unchanged.
Any given object we encounter is really just an "actor," playing a "character" in our lives. It is a physical representation of the mental concept we have of the object. The better we know this object, the more defined this concept becomes, and the more rigid our demands from the concept's physical manifestation (the object itself).
Changing, removing, or substituting materials is similar to changing the actor's hairstyle, wardrobe, or even the entire actor. But as long as the concept of the original object (character) remains the same, and nothing about the new or missing material (actor) directly undermines that concept, the object's "character" remains unchanged.
And, as with the Darrins, mileage varies from person to person. To some, adding an outboard motor to Theseus's ship wouldn't change the fact that it's still Theseus's ship. Some would be unhappy if you replaced a single nail. It comes back to how familiar one is with the original, and how rigid their concept of that object's "character" has become.
So, going back to Asura's Wrath: If a person's concept of the game focused primarily on the story, a gameplay change could be forgiven. If a person's concept of the game relied more heavily on the mechanical experience of play, not so much. This leaves us an answer that's a bit of a non-answer:
Would it still be the same game? To some.