Is a non-correlating signed identity proof possible?

Is it possible to receive from a party $I$ a signed message $M$ that can be presented to two independent other parties $V_1$ and $V_2$ as $M'_1$ and $M'_2$ in such a way that they cannot establish a correlation between $M'_1$ and $M'_2$, but that they can verify that the message was signed by $I$ and issued to the one presenting it?

EDIT:

To make it more clear what I would like to achieve here's an example:

The message $M$ contains the statement "the owner of public key $PK_H$ has blue eyes". The owner of the matching secret key $SK_H$ to $PK_H$ can prove that this attestation was issued to her. So everyone who trusts $I$ trusts that the person presenting that message and can prove knowledge of $SK_H$ has blue eyes.

This message is then sent to $V_1$ and $V_2$ so they can verify if the sender has blue eyes. But to prevent that $V_1$ and $V_2$ can collaborate and figure out that this is the same person, the message must be presented to them in some derived form, therefore $M'_1$ and $M'_2$.

It might well be completely impossible, but I'd like to confirm that.

Yes, you can do it, instead of sharing the signature of the message (msg='alice has blue eyes'), you will generate a zero-knowledge proof that will prove that you have a valid signature of that message, and only share the proof with the verifier.

The proof will be different for every verifier as the verifier will ask you to embed a random challenge into the proof, so the prover will need to generate a fresh proof for every verifier that request the same verification, which will result in a different proof every time and that will enforce the un-trackability between the different interactions with the different verifiers.

If you want to read more about it the topic is called "Non-correlating signatures", here is an example of a paper that mentioned it and discussed the zkp solution - zkKYC