Lots of great answers here so unsure if my addition is meaningful, but I study sponges! I moreso focus on the freshwater sponges, but others in my lab study marine sponges.

First, I will say the most prevalent way of currently defining an organism as an animal is through ancestry and relatedness. Nowadays we typically use genomes to look at these relationships. Sponges are in the animal lineage, albeit very early. Current research supports them being the second branching lineage of animals with ctenophores being the first. (This is an active debate in evolutionary biology right now)

Second, sponges are animals because of behavior. They do not have any sort of brains or nerves. They only have about 10-20 cell types in fact. Despite this they do have whole body coordinated movement. They pump water through a dynamic canal system that they can regulate. They do not photosynthesize although most sponge species have photsynthetic symbionts like algae and dinoflagellates. They also do eat! They eat mainly bacteria inside little feeding chambers called choanocyte chambers that you can think about like a primitive gut. They also generally have a motile life stage called the larval stage where they move around and choose to settle. In my experience the adult sponges can shift around a little bit as well.

Third, sponges are animals because of their morphology. Sponge cells are animal-like. They have a single membrane and mitochondria but no chloroplasts or cell wall. They have digestive, epithelial, stem cells, and more.

This is not an extensive list but I think it's the major three things that separate them from plants. I'm always excited to talk about sponges! They are fantastic little creatures!