For sympathetic vibrations, the main frequency will be the driving frequency, but non-linearities will also couple the sympathetic vibrations into vibrational modes with other frequencies as well... but these other modes will in general have frequencies that are not multiples of the driving frequency so they will not resonate and will be minimal (ie, much more minimal than in the normal hammer driven modes where there is not a "driving frequency"). Below I describe this in more detail:
For the case where everything is linear, as you know, only the driving frequency will be present (even in a perfect string that has many perfect harmonics, there's just nothing to get any power into those modes).
For the non-linear case, I'll start with an important distinction between continuously driven and impulse driven instruments:
It's common for people to refer to "the fundamental and its harmonics", but as generally used this is a misnomer. The key point in physics that makes this important is that for continuously driven oscillations (like a violin, flute, oboe, etc), the overtones are harmonic, that is exact multiples of the fundamental (because anything that's not a harmonic changes phase relative to the driver and is built up when in phase and diminished when out of phase, ie, not resonant); but in impulse driven instruments (like a piano, guitar, etc) there is nothing forcing the overtones to be harmonic, and they generally aren't.
For a typical hammer-driven piano sound, there are a few non-linear things going on. The most important is the stiffness of the strings, which generally stretches out the overtones (and this is what makes pianos difficult to tune). There are other things going on as well. For example, the typical string vibrations we think of are transverse, but there are also longitudinal modes as well, and the transverse modes couple to and drive longitudinal modes. (Basically, I think the longitudal modes are excited by the stretching of the wire during large transverse deflections.) Because transverse and longitudinal modes have different speeds, there's no intrinsic relationship between their frequencies, and they are inharmonic. These longitudinal modes are perceptible, and sometimes included in synthesis.
So why am I saying that these non-linearities are unimportant for sympathetic resonance? The key point is that sympathetic resonance is being driven by the other vibration at a single frequency, and as with other instruments with driven oscillations, the only vibrations that persist to any degree are those that are continually supported by the driving frequency. (Also, eg, the transverse vibration never become large enough to stretch the string for longitudinal vibration, but anyways, these wouldn't be resonant.) That is, when struct with a hammer, the non-linearities of the string play a critical role in the sound of the piano, whereas the same modes will not be supported in sustained sympathetic vibration because they are not resonant with the driving frequency.
Should you believe this?:
Overall, I've studies this topic (physics of music) but this is predictive based on what I know and you shouldn't believe it 100% (like all predictions). I'm quite confident in all of the supporting info I'm basing this on, but the exact conclusion that other modes aren't relevant, well, I'd like to see and hear some data myself.