You would only capitalize dragon in this instance if it meant something other than "generic name for fire-breathing reptile of fantasy."
For example, in Gregory Maguire's alternate history of Oz (the book series which starts with Wicked), animals which can't talk and aren't sentient are written in lowercase (dog, lion, tiger). However, those who can talk and are sentient, like the timid fellow Dorothy meets in the forest, are called Animals, and get capital letters: Lions, Tigers, Bears (oh my). Dr. Dillamond, who is a professor, is a Goat, not a goat.
In your story, if there's a difference between dragon and Dragon, you might capitalize it. Otherwise, Grammarly is just plain wrong, and you may cheerfully tell it to go stuff itself.