Are there finite objects for which deciding isomorphism is NP-hard or harder?
Graphs and groupgroups are not solutions.
Searching the web didn't return answer for me.
Partial result based on Chow's comment.
From a paper p.16
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.