Possible to determine equivalence of hash codes from different hashing functions?
Actually, if you define the $H_i$ functions properly, it can be done.

I'll make the simplifying assumption that we can treat the $H_i$ and $F$ functions as Oracles (that is, you're not allowed to look inside their implementation); I believe that it's still possible without that assumption (but the solution may be more complex).

For our primitives, we'll use FF1 with a secret key; and define:

$$H_i(x) = FF1_{key}( SHA256(x) || i )$$

This meets all the requirements of a secure hash function (e.g. it is collision resistant if we assume SHA256 is); in addition, the holder of $H_i$ cannot use it as a Oracle to compute $H_j$ for $i \ne j$ (as they don't know the $key$)

And, we can define a function $F$ that uses the secret key to decrypt both encrypted hashes, and compares the SHA256 values.  This will correct tell if the two original messages where the same (unless we happen apon a collision in SHA256)