The source material is crystalline, but it becomes somewhat glassy during the fiber manufacture, when it is cooled too fast to crystallize completely.

Asbestos is made from some silicates (pyroxenes or amphiboles) which contain long covalent chains of silicon and oxygen atoms, which are more likely to separate than to break transversally.

Basalt also contains pyroxenes and amphiboles, but they are mixed with other kinds of silicates and they also have a different chemical composition than those of asbestos, so as far as it is known for now the probability of breaking into very thin fibers is very low for basalt fibers.

It is plausible that basalt fibers should be safer, because unlike with asbestos, which is made from rather rare minerals, basalt covers a large fraction of the surface of the Earth, so if basalt were dangerous erosion should have made harmful basalt fragments abundant in the environment.