If you look at this from a proof-theoretic point of view, then each proof yields certain kinds of information which ideally facilitate the extraction of computable realizers or things in this fashion.

An interesting field of study is the topic of **proof mining** which concerns the extraction of computable realizers or uniform bounds from (possibly non-constructive) proofs. Ulrich Kohlenbach has written an extensive book on the topic [1].

Based on Gödel's Dialectica interpretion and a negative translation of formulas, one can show that if a sentence $\forall \vec{x}\exists\vec{y}A_0(\vec{x},\vec{y})$, where $A_0$ is quantifier-free, can be proven in weakly extensional Peano-arithmetic using only quantifier-free choice and some universal axioms, one can extract realizers (computable functionals) $\vec{t}$ for $\vec{y}$ such that it is constructively provable that $\forall\vec{x}A_0(\vec{x},\vec{t}(\vec{x}))$.

Therefore, it is at least for some cases possible to extract general information from proofs, independent of the actual form of the proof (however, it is important that the axioms used are in a certain set of allowed ones).

NB: When it comes to bound extraction, the quality of the bound might well depend on the actual proof. For example, there are different proofs by Euclid and Euler for the proposition "There are infinitely many prime numbers" which yield different upper bounds on the $(r+1)$th prime number. *I think that this also motivates the construction of different proofs for the same theorems.*

[1] Kohlenbach, Ulrich. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Heidelberg: Springer, 2008.