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How would one discover the Pythagorean theorem without having a proof?
You could draw a right angle triangle and notice that the squares match without being able to prove that.
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Or it was discovered by his followers and students (pythagoreans), or perhaps his name become associated with people who were mathematical philosophers at the time.
Yeah, it could be a case of syncretism: Pythagoras' school was perhaps a meeting of minds that took a bunch of disparate ideas/concepts from all over the world and finally nailed them down into more concrete shapes and merged similar but equivalent concepts into grander shared abstractions. Then things like "Pythagoras's school found" and "Pythagoreans believed" accidentally over the years get rubbed off to read "Pyt…
The early church trying to enforce the slightly odd concept of the trinity found solace in a triangular number being equivalent to god.
As you say, proving this would be practically impossible.