Groot's response to the question would probably be a simple "I am Groot!", but does the evidence suggest that Groot is a plant, an animal, or perhaps a fungus?
After reading this post on the real plants that could have evolved into Groot, a discussion began about what kingdom Groot would really best fit into — plants, animals, fungi, or perhaps even some combination of the three:
Don't ignore his eyes. If he evolved here on earth he would be in the kingdom animalia than kingdom plantae.
Plants can sense light (see phototrophism) so it's not entirely unreasonable that a plant might develop something like eyes. It's also possible that what we assume are eyes do not function as photoreceptors — they just look similar to eyes.
During the dancing clip, Groot does not need to direct his eyes towards Drax to determine when to pause while dancing. This suggests that he has some other way of determining when Drax turns around — possibly because he's not using his 'eyes' to see, he's using something else.
I posit that Groot is not actually a plant at all, but rather a sentient plant virus. As long as a single piece of infected vegetation remains intact, the virus survives. It is able to stimulate rapid growth as well as absorb additional mass from the various vegetation, spores, pollen, and fungi it comes in contact with on the ground and in the air.
Or, given their already complex relationships with plants (often symbiotic) I think a fungus would make a lot of sense and do a lot of the same things you mention here AND offer up an explanation for the bioluminescents. ;)
I've convinced myself that GROOT is an organism like lichen. He is, in fact, a plant AND a fungus.