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Here’s what I wrote in the comments:
Personality doesn’t come with the powers unless the powers are based on personality. However, the cases mentioned here are costumes, and that doesn’t define the personality either. In fact the opposite is true. A character’s costume or armor or whatever is influenced by the personality of the person wearing it, especially if that character designed it in the first place. That’s just good character design.
Someone else mentioned how during the times Dick Grayson took over as Batman he didn’t become Bruce. He kept his personality for the most part. That’s just good character writing.