What could this signal at 119.9 MHz be?
This signal is an extremely inefficient use of bandwidth, as you can see because there is an essentially unused area between the center and ±125 kHz. Therefore, I expect that it is almost certainly unintentional radiation (a.k.a. RFI/EMI) rather than a meaningful transmission.

The origin of the signal could be as follows — there are other ways it could arise but this is a simple plausible one:


There is an oscillator at 120 MHz, a nice round number probably chosen as a clock frequency. (Your 119.9 is either error in your receiver's oscillator or in the transmitting oscillator — it is likely in an application which does not need less than 0.1% error.) This oscillator is not deliberately connected to a transmitting antenna — it is just part of some circuit that isn't designed well enough to not radiate.
That oscillator is being amplitude-modulated by another signal at about 125 kHz (the distance from the carrier to the nearest sideband). This can occur many ways — one of the simplest being if something is varying the load on the common power supply at 125 kHz and the 120 MHz oscillator's output amplitude follows its supply voltage.
The 125 kHz oscillator's frequency is being modulated a bit by something else, causing the visible frequency changes. Again, this is fairly easy to have happen by accident.
Then if we look outward further to twice the frequency, we see a copy with twice the frequency variation but exactly the same shape other than that. Thus, this is just a higher harmonic of the 125 kHz signal. This is more evidence that this is not a deliberate transmission, as this doesn't efficiently add useful information to the signal.