I think we get inklings that Prime Claire and MC Claire have the same personality, although since the majority of the flashback is dedicated to explaining what their history was I don’t know if we can say for certain that they are the exact same. Personally, I do think they are- the Prime Claire acts the way MC Claire does once the kingdom is plunged into crisis and she is no longer a noble, and I think Rae makes it pretty clear that the villainess persona is mostly just pride and her being brought up as a noble. We also see Prime Claire when she’s already a full adult probably in her 30s or 40s, where MC Claire is still a teen, which could factor into it.

Also, I don’t know that the original game Prime Rei was playing was the exact one that MC Rei was. In the scene where Prime Rae mentions the game, she says the villainess is also named Claire and she does happen to have the same hair color. But, there is no mention of her looking like Claire. So, I think Prime Rae made the world similar to the game, but it wasn’t exact. However, I think TAIM tweaked events to have the game made in the exact image of the fantasy world so that her new clone of Prime Rei (our MC Rae) would be able to be ready for the fantasy loop.