Fungi that belong to division basidiomycota are also known as club fungi. They show presence of a stem, a cap, gills beneath the cap and produce basidiospores ( or simply spores). Both mushrooms and shelf fungi show features similar to the fungi of division basidiomycota, such as presence of gills-like structure and fruiting bodies that produce spores.
Hence, the mushrooms and shelf fungi are classified as club fungi or basidiomycota.