In Spanish we have ensayo = attempt or try, frequently is used to refer to attempts before the real show or performance (in a music festival, a theater, ..). Also ensayo is just equivalent to what Montaigned introduced when referred to a text composition.

Similarly, Portuguese's "ensaio", which means "rehearsal".

It also carries the same literary meaning, “ensaio sobre …” == essay, usually only seen in academia, or alternative cinema.

The distinction seems to be different expected outcome for the subject

To "assay" is to engage in an "investigative (analytic) procedure" upon some thing [0].

To "essay" (je vais essayer) is an attempt to achieve a specific thing.

The former seems more like an exploration without initial expectations, like a fact finding, focus group, or initial prototype.

[0] https://en.wikipedia.org/wiki/Assay

That site doesn't seem great, but according to Wiktionary "assay" is a doublet of "essay", i.e. the origin is the same but they were borrowed into English via two different sources.

"Assay" came via Norman, "essay" came via French.

There are lots of other examples like this where one word came from Norman and the other from French, e.g. warden + guardian, warranty + guarantee