My understanding is that Archimedes didn't, and never could have, produced the actual formula, because that way of writing formulas didn't exist. That's discussed in the sub-thread here:
https://twitter.com/daveinstpaul/status/1213469023461138432
Additionally, the result was probably known before Archimedes, but again, writing that formula in that form probably didn't happen until Euler.
> Archimedes discovered the formula many centuries before Euler, although he could not express it in modern algebraic notation.
If Archimedes was able to express the formula in natural language (e.g. "take the radius of the hypersphere, multiply it by itself as many times as there are dimensions, ..." etc.) in a way that was unambiguously understood by other mathematicians (or natural philosophers) of the time, then that formula has definitely been discovered. Writing it in modern notation with one-character symbols for addition, multiplication (typically zero symbols!) etc. is nowhere near required for it to count as being discovered.
It's very easy to take for granted the sheer amount of innovation and ideas that went into being able to express that formula just because they are grade school topics today. The concept and acceptance of irrational numbers, the relationship between geometry and algebra, the use of symbols in algebra and even assigning symbols for special real numbers took centuries and a steady stream of creative leaps of genius.
An excellent and detailed explanation I highly recommend can be found here: https://hsm.stackexchange.com/a/1872.