Mechanical systems have nearly infinite degrees of freedom -- a straight bar in free space, for example, has not only a 1st order flexural mode, but 3rd, 5th, etc. (or other harmonics depending on fixed points), and that's flex in two dimensions (flexing along the longitudinal axis). And there's stretching modes, not just longitudinally but across as well. And torsion modes. And surface acoustic modes, both compressive and shear. And...
In short, mechanical systems are
really, really rich in terms of wave mechanics!
We have a mere taste of this in electronics, with transmission lines (single propagation mode, harmonic resonant modes), waveguides and fiber optics (same things, really) (multiple modes above the first), and free waves (which really just do whatever they do until they get to an antenna, where it goes back down to one-dimensional transmission line). And EM waves are transverse propagation mode only, with the two polarization axes.
Mechanics also matter at much more accessible frequencies. The speed of light is quite fast indeed, so that we can build relatively fast circuits (10s, 100s MHz) without even having to worry about transmission line effects; whereas the speed of sound in most materials is so low that even, say, a car engine running at 100s Hz must be, well---you can't really do much of value from first principles, you have to model and test and evolve the design of the damned thing to get anywhere.
Regarding frequencies, mode conversion isn't at all uncommon in mechanics. While this cannot happen in a linear material, there are many nonlinear behaviors -- flexing an oscillating beam for example, modulates the rate of the oscillation. (Consider the bowed saw instrument.) Harmonic generation and mixing is common through various nonlinear flextures, and rolling and colliding contacts, and so on.
Anyway, as for multi-frequency oscillators -- given this, given that we are much more limited in the ways we can make our electrons move -- there are still ways (of course).
The history goes back to I think early Western Electric DTMF sets, which used a single transistor (they were expensive back then!) and a keypad matrixed selection of LC tanks, carefully tuned for the frequencies required.
The trick is to allow both oscillation modes to persist, without one dominating the other, or the combination resulting in squegging or something else like that.
The problem with the normal oscillator circuit is, the amplitude of the amplifier itself is the limiting factor. Which will mix both modes making IMD, and whichever one has most gain will win out.
If we take a theoretical step back and consider the fundamentals of an oscillator, we might stand to synthesize a circuit based on it. If we take a normal (lossy) tank, and connect a negative resistance across it, we get an oscillator (amplitude grows without bound). If we put a nonlinear element across the tank, so that the amplitude grows to a limit, then it will stabilize. As long as the negative resistance remains negative, we could even put more (parallel) tanks in series with it (or more series-resonant tanks in parallel, same thing), and they'll all do their own thing independently.

So the trick is to find:
1. A component which limits tank amplitude, and
2. A negative resistance network, or amplifier, that remains reasonably linear over the total amplitude and bandwidth required.
Note we need a limiter that acts without severely affecting the tank itself -- a simple clamp will affect the frequency, and the oscillator will "chirp" undesirably.
The answer isn't very complicated (a great relief, as these theoretical explorations can often go very differently!). A negative resistance, over a reasonable range of amplitude and bandwidth, can be made with a single transistor, a transformer, and some assorted resistors and capacitors for biasing and setting feedback gain.
The limiter can be something like a MOV, perhaps with some series resistance to soften its detuning effect. (How much resistance can be afforded? Well, not more than the negative resistance, that's for sure!) A lamp (as seen in many Wien bridge oscillators) probably can't be used, because it will respond quite slowly. Or maybe it'll work out in the end, but the startup transient will have undesirable AM on it, as the two modes fight it out initially.
With an op-amp, simply use an impedance converter circuit, then connect it to a pair of LC networks, each of which has a limiter on it (which might be back-to-back LEDs with some series resistance, say).
Tim