Nope, definitely a sponge. The structure you circled is the osculum, where the water flows out of the sponge.
Unfortunately, you'll probably have to control it by hand. While many sponges are sensitive to air exposure, not all sponges are. Shipping and acclimation in the hobby tends to weed out the species that are sensitive to air exposure so what we get as hitchhikers are the ones that tolerate air exposure well.