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Ingo being an established character from the setting's present day with partial amnesia = Just fine! No one questions it. But Lucas/Dawn being an established character from the setting's present day with partial amnesia = preposterous? Because it's not made painfully explicit.
The real reason it's framed the way it is is because of character customization, OP. By default your character is the same hero that dealt with Giratina and Team Galactic in the present, sent to the past by Arceus. Which is why your character is treated as some great prodigy. Given what we know it's safe to assume this, rather than pretend it's just a coincidence that the default hero looks exactly like that character we know.
But if you want to customize your character and pretend like it's not related to a previous game you might not have even played, the details are vague enough to let you pretend otherwise. People (retards) take this and try to pretend like that proves the opposite: That these characters CAN'T be their counterparts from the present sent back in time. They use the ambiguity they were given and claim that it proves there's no ambiguity at all.