Depending on mechanical limitations, Chinese and Russian illuminators can do it though, although both countries don't actually use a "pure" illuminator like the SPG-62.
That's because these so called "illuminators" appear to be as true radars, capable of tracking and generating tactical range information. For the operators that would mean a display where the radars could not only track heading, but display range information and therefore location.
Not just ship fire control radars, but also land fire control radars.
These radars would be implementing FMCW, if separate transmitting and receiving arrays are observed. Or they may be implementing ICW or FMICW, if a single array is observed, where interruption is used to produce send and receive time sharing on the same array.
However, I think they are using this sort of ICW, as opposed to what Ambi is describing. These are not pure illumination devices, but true, self contained radars.
You also don't need to change frequency. Just change the frequency modulation for different missiles, so each uses a different modulated wave form for reference.