Tallow and oil base shaving soaps are what most shaving soap makers are producing nowadays. Oils like palm, coconut and nut oils bring unique properties to a shaving soap, most soap makers also add butters like Shea butter or Olive butter to achieve the desired properties and qualities. In some cases the actual amount of tallow has been reduced so much I'd personally not call it a tallow base base soap anymore, but as the word 'tallow' seems to be golden standard for shaving soap makers are reluctant to name their soap bases to something more reflective of the actual ingredients. Using butters and oils to make soap is generally easier than using tallow as rendering the tallow is quite a task in itself, but the use of expensive oils and butters could push pricing of these to the same or beyond that of tallow soaps. Soaps can also contain other animal products like goats milk or lanolin.
Tallow and oil base soaps are generally medium to soft depending on the recipe and maker, these soaps can NOT be melted down to another shape or form and must be manipulated by grating and/or pressing it into a form using your thumbs and medium pressure.