“Middle class” does not map to the bourgeoisie. The whole lower/middle/upper class distinction is so poorly defined that you can say literally anyone is middle class. Might be upper middle class or lower middle class, but you’re never not middle class. Worker/bourgeoisie/nobility is more specific, and it’s a trap to try to see it as directly mapping over.

The term bourgeoisie was a distinction from the nobility who owned the land by right of birth. At the time, there was no other way to own land by purchasing it on the open market. The bourgeoisie emerged with early capitalism, and wanted to own land. The American and French revolutions were largely driven by the bourgeoisie with the help of the workers.

In chapter I of the Communist Manifesto, the bourgeoisie are the heroes. They did break the power of the nobility’s stranglehold on land ownership. Right-libertarians sometimes use this fact to say capitalism is great, but this is the wrong way to look at it. Marx and Engles never argued things should stay there just because anyone can theoretically aquire enough wealth to own land.