It is difficult to comment on exactly what the college student in the TV show was doing, since there are very many different types of spider silks, sponges, and uses for them. However, the general idea of using spider silk for various speciality sponges is definitely possible.
Sponges have to absorb a large amount of water, or other liquid, and can be made out of many materials - the only requirement is that they do not repel water (hydrophilicity), they can stretch when absorbing the water (elasticity), and they have many pores. Of course, depending on what the sponge is used for there can be other requirements. Any hydrophilic, elastic material can be made into a porous structure that will work as a sponge.
Spider silk is a very unique, expensive material, so it would make only sense to make a sponge out of it if there were some very specific requirements that could only be satisfied by the spider silk.
Sponges you would find in a kitchen for washing dishes are usually made of common plastics like polyester or polyurethane which are stretchable and can be made to absorb water. More environmentally friendly, biodegradable sponges can be made out of cellulose (wood. These materials are cheap and strong enough for a dish sponge, so there is no particular need for something as exotic and complicated as spider silk.
However, sponges can also be used for much more complex tasks, such as gradually and controllably releasing medicine in a specific region of your body. In this case, the sponge needs to be safe to be eaten or injected (biocompatible), degradable in your body (biodegradable), and tunable to work with a specific medicine. Some spider silks work uniquely well for this purpose - they do not cause any bad reactions in your body, degrade naturally, and can be genetically engineered to work precisely with specific medicine. In this case, the spider silk is a great material, and its cost is not a problem since only a small amount of it is needed for a very specialized, complicated task.
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