I think you can make a philosophical argument that a fully formal proof, where every claim is traced all the way back to the axioms, is not a social construct.

But when we say "proof", we usually don't mean "fully formal proof", and there are good reasons why:

1. Fully formal proofs didn't exist for most of the history of mathematics. 2. School by and large don't teach formal proofs, and most students are probably not aware of the existence of formal proofs. 3. Even today, most professional mathematicians never write formal proofs, except perhaps as an exercise during their education.

So what do we mean by "proof" if it is not an argument that is exhaustively traced back to axioms? It's really no more than "that which the teacher/other mathematicians will accept as a convincing argument". When you learned a proof of the Theorem of Pythagoras in high school, you almost certainly didn't learn a fully formal proof. You learned a proof that, at some point, just implicitly got "cut off": the proof tree didn't work all the way back to the axioms but stopped at some point where the argument would become too tedious to continue laying out in full (without even being told that the proof got cut off: your teachers perhaps even told you that this was a rigorous argument).

To write such a proof, you need to judge where the acceptable cut-off point is, which is entirely based on what other people will accept as good work. Hence, a social construct.

edit: if you're not convinced that the proofs you learn in school/university aren't fully rigorous, I warmly recommend trying out a proof assistant like Coq, Agda, or Lean. Try to encode some well-known piece of mathematics. Euclid's Elements is a good candidate: working through it fully formally, you'll find huge omissions in the Elements immediately.