Proving equality of two Schnorr Signatures
I don't think that works with keeping the message private - unless you break the signature scheme by reusing the nonce.
Your variables $c_1$ and $c_2$ are the results of the hash function. And those can not be the same by the definition of $r$ being a nonce. Moreover: the hash function does not retain any algebraic structure. By the definition of hash functions, the resulting hash only shows if two inputs wre the same or not. No partial matching or any type of relation between two inputs.
But they are also the only variables, that depend on the private message $x$. And  from that, I don't see any way to distinguish $c$-values , where $x$ is the same or $x$ is different. Unless you allow $r_1=r_2$, which breaks the security of the signature.
The main misconception with using Chaum-Pedersen: They don't use signature schemes. They use commitments. From the statement, that you want to keep $x$ private, maybe you should use commitments instead of digital signatures. They give the privacy requirement, and you can use Chaum-Pedersen directly.