What is the mathematical meaning of the plus sign (+) in chemical reaction equations?
I think we are over-interpreting the poor $+$ sign. In older books you will also see an $=$ sign instead of an arrow just like what Lavoisier did. We have to get hold of La Traité Élémentaire de la Chimie (1789).
We should look at the historical perspective and I always want to see the earliest usage in chemistry just to see what was the original person thinking. I think we can credit Lavoisier for the &quot;+&quot; sign in chemical equations and let us see what was going on in his mind. I quote from the History of Chemistry, by F.J. Moore 1910 (Full text in Google Books).

The cumulative effect of these researches led to the sudden conversion of most French chemists about $1785,$ and the new ideas were firmly fixed by the publication of Lavoisier's great text-book La Traité Élémentaire de la Chimie in 1789. The change was well called the Chemical Revolution, for it inverted completely the chemical point of view. The mysterious hypothetical substance, phlogiston, which did not obey the law of gravitation, and changed its properties arbitrarily as theoretical considerations dictated, was banished from the science and the law of conservation of mass vindicated once for all. This made possible quantitative analysis and the chemical equation. The latter is a convenience in form of expression which we owe to Lavoisier, and of whose assistance in stating and solving chemical problems he made immediate use. He writes: $^{1}$
If I distill an unknown salt with vitriolic acid and find nitric acid
in the receiver and vitriolated tartar in the residue, I conclude that
the original salt was nitre, and I reach this conclusion by mentally
writing the following equation based upon the supposition that the
total weight is the same before and after the operation.
&quot;If $x$ is the acid of the unknown salt, and $y$ is the unknown base
$I$ write: $x+y+$ vitriolic acid $=$ nitric acid $+$ vitriolated
tartar $=$ nitric acid + vitriolic acid + fixed alkali. Hence I conclude: $x=$ nitric acid, $y=$ fixed alkali, and the original salt was nitre.&quot;
Nothing like this had ever appeared before in chemical literature and
what a fog of mystery and superstition it removed!

Update Surprisingly, I cannot find this quote in Lavoisier' book (can't read French) but + sign is obvious! The only equation in his 382-paged book is
moût de raisin = acide carbonique +  alkool.
grape wort = carbonic acid + alcohol
Anyway, plus sign is quite old. In the book titled Historical studies in the language of chemistry by Crosland, Maurice P, a 1744 text is cited showing

Readers should consult this book (Internet Archive, need to make an account for borrowing it) if the want to dig deeper. The author discusses plus sign at several places at length.