Soaps that don't contain tallow are usually referred to as "glycerin based" and can be melted and poured into a container, such as a shaving mug or plastic storage container. It's usually done in a microwave.
"Melt-and-pour" is the correct term for the "soap" that is incorrectly called "glycerin-based". The reason it has the name "melt-and-pour" is because all it involves is melting a base and pouring it into a mold. After you have melted the base, you can add fragrance, glycerin, colorants, etc. Brambleberry sells a shaving melt-and-pour base here.
Just because a soap does not contain tallow does not mean it is melt-and-pour. And you can most definitely melt a soap with tallow in it.