There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the (Euclidean geometry) axioms. Are all these proofs equivalent? Do they all track back to the same axioms?
Sure, the Pythagorean theorem is an item in the theory of Euclidean geometry, and it can be derived from the modern axioms of Euclidean geometry.
A full set of Euclidean geometry axioms contains the information about similarity and area that are sufficient to prove the Pythagorean theorem "synthetically," that is, directly from the axioms. The algebraic proofs are a little different, though!
It turns out that after defining the real numbers and basic algebra, you can create a model of Euclidean geometry in $\Bbb R\times \Bbb R$ which obeys all the Euclidean axioms. The algebraic operations in an algebraic proof reflect the synthetic axioms being used, but the direct connections are not obvious. You are still indirectly using the synthetic axioms, but they are all hidden assumptions about $\Bbb R\times\Bbb R$ and coordinate geometry.
Your Question is much more complicated than it looks.
First some philosophy:
Are all proof equivalent?
What do you mean by equivalent in relation to proofs? (not just in relation to this proof but to any proof at all)
Euclid was a bit lacks with his Postulates and Common Notions, The axioms of his geometry were only found in late 19 , beginning 20 century so it is reasonably fresh.
see for example: (there are many more and even within these examples there are different options)
PS: Make sure you use the axioms for Euclidean geometry, you need to add the parallel axiom or an axiom that (together with the other axioms) can proof it.
Theories (and Euclidean geometry is a theory) are defined by their theorems (everything that follows from the axioms and rules of inference) not by their axioms, so many different axiomatisations can give the same Theory. But for being the same theory they have to proof the same theorems.
Then the Pythagorean Theorem.
Yes you can proof the Pythagorean Theorem in any axiomatisation, if anything if you could not prove it it would not be Euclidean geometry.
Do they all track back to the same axioms?
No, there are different axiomatisations possible so they are not even able to track all back to on and the same set.
Hope it all helps
All proofs I have seen so far trackback to the parallel axiom, or the axioms of similarity(which I think are equivalent). I'd be interested to see these "algebraic" proof though. Anything proved by pure algebra alone must be universally true, which is not so in the case of the Pythagorean theorem. It must use (euclidean) geometry in some form or the other.