Octopuses have 8 tentacles, spiders have 8 legs.
Is there something special about 8? It seems like an animal that needs 360° mobility has 8 legs.
No, there is nothing special about eight legs. You have species with two legs (humans), four legs (most mammals), six legs (insects), eight legs (spiders), ten legs (lobsters), and many more legs (millipedes, centipedes).
This is just a wild guess, but for movement on a surface in which legs are lifted from the surface, six legs might be sufficient to provide three points of support at all times (think three-legged stool) while allowing relatively rapid movement (legs on both sides can be moved simultaneously), but such might be both less stable and less friendly to variable direction of movement (since legs are in fixed locations and not evenly/radially distributed in animals with bilateral symmetry, having more legs would facilitate balance for different directions of movement). Having eight legs allows well distributed support and might allow more simplified motor control.
Having more than eight legs would tend to lengthen the body (for animals with bilateral symmetry) which might tend to bias movement direction (as well as possibly being wasteful of resources).
There might also be correlation of highly variable movement direction with requirements for greater stabilization. E.g., eight legs might provide greater stability on a web (which surface might encourage more variable movement directions) and better support large claws (which might make sideways movement more useful in face-to-face encounters).
Note that the above does not address non-walking animals nor animals with radial symmetry (For an animal with radial symmetry, six legs might be sufficient to provide support with three.).
Hopefully, a better answer will be given, since I have not even paid attention to how spiders (or crabs) move. (The manner of movement would give insight into how significant support/stability and motor control are.)