Validating that a hash is over a specific value for two hashes with additional (known but different) input $M$ (not public) is hashed with a hash function $\text{H}$ that includes a variable $T_1$ (public), giving $C_1$ (public). The same $M$ (not public) is hashed with the same hash function with another variable $T_2$ (public), giving $C_2$ (public).

Knowing the variables, $C_2$ can be verified to be from the same data as $C_1$.

Is there a hash function $\text{H}(M + T_1) = C_1$ that relates to $\text{H}(M + T_2) = C_2$ in such a way that, knowing $T_1, T_2, C_1, C_2$, and with no knowledge of $M$, it can be verified whether both use the same $M$?
This is not probably what you had in mind; but this is easy to achieve with a slight expansion.

Let us assume we have a private, public pair, with private key $P$, and public $K$. We basically consider $P$ to your $M$, and take $K$ to be part of the public knowledge of the verification of the hash.

Then $H(M+X)$ is defined to be $\textrm{Sign}(P,\textrm{Hash}(X))$ for any desired hash algorithm. Let $C_i=\textrm{Sign}(P, \textrm{Hash}(T_i))$. Given the public $K$, one can then verify the legality of each $C_i$ (and of course, for publicly known hash-function, one can also calculate the hash of $T_i$ directly).