Q about sugar

So actual Sugar cane and the sugars occuring in (other) plants are different things, though it obviously contains plant sugars, but why is Sugar cane called sugar? Or i guess sometimes they use beet Sugar in “sugar” and call it “sugar”. What is it in sugar cane and beet sugar that’s comparable to the sugars in other plants, to where it would be called “sugar”? Is it mostly fructose, or sucrose, or what? Is plant sugar a lectin? I’ve seen that it’s mentioned as a type of plant defense. These words are often used interchangeably and I think it causes some confusion in the medical field.