We have had pretty dumb arguments in the past. Mostly because I could never accept another person's opinion. I see where you are coming from on this though.
I agree you can't have a zanpaktou that isn't your zanpaktou.
However, Rukia gave him Shinigami powers. Lets discuss this. Zangetsu was still inside of him, but he still had a different sword.
The huge Berserk esque "zanpaktou" from beginning of manga was a "shikai that was not Zangetsu." You know what that first sword probably was? A fullbring. Just like the sword he had after FGT. That was probably a fullbring. Zangetsu and his Hollow side are gone. When he received "shinigami" powers again, I would think it was similar to the first occurrence. He received reiatsu, and Fullbring'd a new Zanpaktou. He's only known battle as a Shinigami rep. He wouldn't create any quincy weapons, he knows 0 Kido. In fact, the only thing he knows how to do is swing a sword.
The zanpaktou that Ichigo currently has is NOT Zangetsu. Link me where Ichigo recently goes bankai and calls Zangetsu by name after FGT. Its just fullbring. He thinks he still has him, so he "Bankais" which is just another Fullbring. Fullbringers get kicked the **** out of Soul King's palace. Zangetsu could not be repaired, because he is long gone. Its over for Ichigo.
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^That is not Zangetsu Shikai.
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^This is where he went bankai against Kirge
No mention of Zangetsu by name.
That overly huge zanpaktou that Ichigo had after Rukia gave him Shinigami power
was Ichigo's true zanpaktou. It was just unreleased. In fact, that sword is the
same sword that Ichigo gets after he retrieves his Shinigami powers a second time with Urahara.
This is the original sword Ichigo has in the beginning of the series:
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It's also the same sword that Byakuya cuts in half during their first encounter:
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That broken blade is the same blade Ichigo recovers at Urahara's place after he helps him gain his Shinigami powers again. Note that it's still broken from his encounter with Byakuya:
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Urahara busts it further:
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Finally, this is when Ichigo releases Zangetsu for the first time. The full-time released Zangetsu that we've all come to know came from that broken hilt - the same zanpaktou from the beginning of the series:
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(Ignore the horrendous translation error)
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My statement from the beginning was only that Ichigo's zanpaktou is Zangetsu. This is of course only under the assumption that whatever Ichigo has right now
is a zanpaktou.
As for the question of whether or not Ichigo's sword is actually a zanpaktou, the only clue we have is Isshin. When Isshin was teaching Ichigo how to attain the Final Getsuga Tenshou he seemed to imply that he'd done it before:
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To me it sounds like Isshin did something similar to the FGT before and regained his Shinigami powers. If this is true, then it's not out of the question for Ichigo to have done the same. Meaning that it's possible that Zangetsu (and his Hollow) aren't gone forever. Though I admit this doesn't rule out the possibility that Ichigo's sword could be something different