Euglena do photosynthesis using the same basic process that plants use. They also move around and eat, as do animals. They are not classified as either of these. That is because they are unicellular. “Uni” means one (a unicycle has one wheel). In order to be classified as a plant or animal, an organism has to be multicellular, or made of more than one cell. Since it is a unicellular organism with some plant and animal characteristics, it is called a protist. Plant cells have walls. There’s no cell wall around a Euglena’s cell membrane, so it is a protozoan. It used to belong to the kingdom Protista. Now most scientists think we should stop using that kingdom.
Keep in mind that humans make classifications in order to organize the amazing complexity of the world. The natural world does not fit neatly into all of our classifications. We can be pretty sure about how closely related things are by analyzing their DNA, but there’s no magic number that means they belong in the same kingdom, order, or family.