What is this spongy material?

It is some kind of soft-open-cell foam, wrapped in conductive tape, to ensure a good grounding and shielding contact between pieces. The foam may or may not have large amounts of carbon particles in it to make the foam itself conductive.

As established this is an EMC foam gasket. It is commonly applied to lids of stainless steel enclosures and used for various grounding practices to improve ground connections or to provide a faraday cage for the blocking of electromagnetic fields.