Question

What are the similarities that plant-like, animal-like, and fungus-like protists share?

Answer 1

Well, they're all protists, but that group is nowhere near monophyletic.

Some are actually more closely related to animals than they are to other protists! And even many of the "plant-like" protists are quite distantly related to each other.

Dinoflagellates (plant-like protists), for example, are far more closely related to Ciliates and Apicomplexans (animal-like, I suppose... apicomplexans are parasites and are unusual in that they used to be plant-like: their ancestors had chloroplasts) than they are to other plant-like protists or to plants themselves.