Are there finite objects for which deciding isomorphism
is NP-hard or harder?

Graphs and groups are not solutions.

Searching the web didn't return answer for me.

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Partial result based on Chow's comment.

From a [paper p.16](http://web.cecs.pdx.edu/~mperkows/PerkowskiGoogle/drechselr-10030112.pdf)

> for IBDDs (Indexed BDDs) ... the equivalence test is coNP-complete even
if there are only two layers.

Indexed BDD are rooted digraphs, representing boolean function.

"Equivalence" appears very close to isomorphism and means representing the same boolean function.