There are different theories. The most convincing one is it came from the occitan language where "es per lo et" would translate litteraly in english to "it is for the and". Sounds similar to the origin of ampersand.
The letters which were also words were prefixed with "per se" (it was not bracketed like MW says "I per se I" - why?).
So, when schoolchildren recited the alphabet, they'd say
"per se A, B, C, D, E, F, G, H, per se I, ...".
Now, the "&" was at the time considered the last letter, and it was introduced by "and", as the last item of a list often is ("Peter, Paul and Mary"), thus the schoolchildren would end
"... W, X, Y, Z, and per se &." [1]
French, similarly, ended in "X, Y, Z, et per lui &."
That seems to me a very plausible theory for ampersand and esperluette.
Note, however, that the English Wikipedia mirrors the "A per se A" explanation from MW.
[1] https://de.wikipedia.org/wiki/Et-Zeichen#Englische_und_franz...
https://fr.wikipedia.org/wiki/Esperluette#%C3%89tymologie
> « et, per se, et » (« et, en soi, 'et' ») pronounced « ète-per sé-ète », would have transformed into « et, per lui, et »