Is it possible to derive a constant value from different signatures of a private key? For example perhaps deriving the same PRF from RSA signatures on different messages from a (not known) private key?
Struggling to find the proof of why not.
I understand the question to mean: is there a function $F$ whose domain is signatures under a public-key signature scheme such that, given a signature $s_1$ made with a key $k_1$ and a signature $s_2$ made with a key $k_2$, $F(s_1) = f(s_2)$ if and only if $k_1 = k_2$? Or in simple terms: can you tell who made a signature by looking at it?
You aren't going to find a generic proof of impossibility because it is possible to build signature schemes for which $F$ exists. In fact, any signature scheme can be adjusted to make $F$ possible. Given a signature scheme whose signature function is $S$, define a new scheme whose signature function is $S'(k_{\mathrm{priv}}, x) = (S'(k_{\mathsf{priv}}, x), k_{\mathsf{pub}})$ where $k_{\mathsf{pub}}$ is the public key associated with $k_{\mathsf{priv}}$. In simple terms: attach the public key to the signature. This new scheme is just as secure as the original since the new signature function doesn't reveal any more information than the original and doesn't have equality cases that the original doesn't have.
For ECDSA, given a curve with the generator $G$, its order $n$, and two key pairs on this curve $(d_1, Q_1)$ and $(d_2, Q_2)$, here's a way to construct signatures that are valid with both keys. Note that they are signatures of different messages. Given a nonce $k$ and a message $m_1$, the signature under the first key is $(r, s)$ where $r$ is the x-coordinate of $k G$ modulo $n$ and $s = k^{-1} (H(m_1) + r d_1)$. Let $e_2 = k s - r d_2$. If there exists $m_2$ such that $H(m_2) = e_2$ then $(r,s)$ is the signature of $m_2$ under the second key. Hash functions are not necessarily surjective, so $m_2$ might not exist, but if $H$ is sufficiently indistinguishable from a random oracle, the proportion of values of $k$ such that $m_2$ exists is the size of the image of $H$ among bit-strings of the appropriate length, so many values of $k$ are suitable.
Note that this is a (sketch of a) proof of existence. Finding actual examples (at least with this method) is infeasible since it requires calculating a pre-image under the hash function.
I haven't worked out the math for RSA. I think that due to the longer signatures, it is not the case in general that given two key pairs, there exists a pair of messages such that the signature of the first message with the first key is equal to the signature of the second message with the second key. I expect that this is true with high probability when the hash is sufficiently longer than half the key and false with high probability when the hash is sufficiently smaller than half the key.
Note that if you require the signatures to be of the same message, the answer is completely different. With a known message, ECDSA allows key recovery. This is not true for RSA.
A more interesting version of this question in practice is, given a signature, or more generally a set of signatures made with the same key, and a collection of public keys, is it possible to tell with a non-negligible probability which key made this or these signature? This is the problem of key privacy for signature. RSA does not have key privacy. Ed25519 looks like it might have key privacy.