Multiple windings on single ferrite ring vs multiple ferrite rings
Multiple turns increase the flux density in the core by a factor of N and increases the inductance by a factor of \$N^2\$.
Increasing the number of cores increases the inductance by a factor of n, while leaving the flux density in each core the same.
This the energy stored in the field without saturating the core is a constant per number of cores.
You appear to have a common-mode choke arrangement, so it is vital that all the wires go through each core the same number of turns. Running (say) one wire through each core would be a far less effective use of the cores.