Did the people of the past consider plants alive? If so, how did they come to the idea that plants have something in common with animals?
You do not need to look further than Aristotle. In On the soul, he considered plants to have a "soul" (we would perhaps instead call it "essence") which made them capable of two things: reproduction and growth. The "souls" of animals also gave them the power to sense the world and move in it, and the human "soul" gave rational thought.