As far as I know, a general answer depends on protocol under analysis (as Schnorr) being a Proof of Knowledge (PoK) and not necessarily being a Sigma Protocol.
PRELIMINARIES
A Knowledge Extractor (KE) exists, implied by the protocol being a PoK, and roughly defined as:
an entity capable -outside the constraints of proof execution- of
extracting the Witness W of the Prover’s knowledge, only $\forall$Prover* s.t.
$P [$Verifier is convinced$]$ > $\eta$
where Prover* is a Prover with ANY strategy (so also a cheating one, not necessarily the one prescribed by the protocol).
It seems reasonable to define $\eta$ as "KE error", a threshold below which KE cannot extract W.
THESIS
The soundness error is $=\eta$
PROOF
KE extract W $\Longrightarrow$ statement is TRUE (because W is an evidence of the protocol's statement)
taking the contrapositive:
statement is FALSE $\Longrightarrow$ KE never extracts W
but from KE definition:
KE never extracts W $\Longrightarrow$ $\forall$Prover* $P [$Verifier is convinced$]$ $\leq$ $\eta$
chaining the two implications:
statement is FALSE $\Longrightarrow$ $\forall$Prover* $P [$Verifier is convinced$]$ $\leq$ $\eta$
which is exactly the soundness definition
CONCLUSIVE REMARKS
if $\eta$ = 0 we get perfect soundness , and $\eta$ < 1/2 leads to protocol statistical soundness by protocol repetition and majority voting ; when $\eta \geq$ 1/2 we are in the quite common case in which a satisfying PoK is obtained only by $n$ sequential repetitions of the original one: the resulting protocol can be proved to have KE Error = $\eta^n$, permitting again statistical soundness for a large enough $n$.
If you need more context you could try this: https://github.com/baro77/ZKbasicsCS (mine) or a lot of much better resources out there.
Hope I have helped you a bit
