My understanding is that basalt fibers seem to be glassy, not crystalline, so the breaking does not happen.
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.
While there are glasses much stronger than ordinary glass, there are a lot of even stronger ceramics, which are (poly-)crystalline.
Glasses have many advantages vs. other materials, e.g. easy processing for making any shapes, including fibers, no porosity, chemical resistance, optical transparency and so on, but strength is not one of them.
The glass content of the basalt fibers is useful for allowing them to be drawn into fibers, by being soft enough for this even at a temperature under the melting point of the basalt.
Is there a reference for that, because it's curious. (As in I really hope you have a reference to read.)
I'd thought I had seen that repeated scarring (from being not broken down) upregulates cellular replacement rate, or concomitant inflammation were suspect.
[1] https://archive.cdc.gov/www_atsdr_cdc_gov/csem/asbestos/how_...