Inductors don't have voltage rating, but transformers do... why is that? Why not just put a current, frequency, and turns ratio for transformers and be done?
There is a voltage rating and this is the insulation voltage breakdown of the enamel on the wires. Good inductors should state this somewhere in their data sheet. This applies equally to transformers and inductors.
There is also another implied rating that is linked to flux density and core saturation - basically it's called volt-seconds (aka magnetic flux) and relates mathematically to how much total flux is produced by 1 turn (no matter what dimensions) after one second having had a constant 1 volt DC applied to coil terminals. Given the importance of core saturation there is an implied volt-second rating for any inductor of a certain cross sectional area and number of turns.
The reverse is true - if you linearly collapse a field of 1 weber in 1 second, any coil that totally links to that collapsing flux would have induced 1 volt DC across its terminals.
Why not just put a current, frequency, and turns ratio for transformers and be done?
What about wideband transformers such as audio transformers or Rf transformers - this requires that a band of frequencies be known and not just a frequency.
As for current, there are two currents of interest in a transformer. There is the load current and there is the magnetization current. Both are largely independent and most reputable manufacturers will provide this information.
For a flyback power supply transformer it is more relevant to specify primary and secondary inductances. From those values turns ratio can be inferred but, this is of little consequence for flyback designs, because the inductance values are more important.