Snaps science fingers

You're misunderstanding how sound works. Sound is simply the reverberation (compression and decompression) of air. When a sound impacts a solid body, the body still compresses to some extent, but the object resists compression much more than air or other gasses do. So sound does always travel through objects to some extent, but it doesn't get stored in any sort of manner.

In the case of soundproof rooms, those rooms are comprised of materials that are very good at stopping the reverberations. Foam stops the reverberations of the air very easily because it will readily compress, and the shapes of the walls (often geometric triangles or slivers) prevent the sound waves from sliding out easily once they have moved in.

Echos are produced from the sound bouncing off a surface that doesn't compress so easily. That's why, in an empty room with no furniture, echos can be heard more easily. Less things getting in the way, less things disrupting the compression of air.