Numbers with similar powers

Husk, 10 bytes


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§ḟ       →     Find the first number starting from the input + 1 such that...
        p        The prime factorisation
       g         with equal elements grouped together
    ȯÖL          and sorted by length of groups
  ¤≡             has the same shape as the above applied to the input.

Mathematica, 61 bytes


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-4 bytes from @Misha Lavrov

Jelly, 15 14 bytes

1 byte thanks to Erik the Outgolfer.


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