FM radio signals are not Additive. If you add two FM signals together, you do not get one "mixed" FM signal. You just get noise.
Fortunately, FM receivers are good at picking an FM signal out of noise, so your radio is able to separate one - or the other - of the two signals out of the noise. It does this by locking onto the phase of one carrier signal, which allows it to treat the out-of-phase signal as amplitude noise, which it always rejects.
The problem is that, as the two signals drift in and out of phase (something that happens because of the music, even if the two transmitters are perfectly phase locked, which they aren't), the locked signal gets captured by one - or the other - of the two transmitters. Which signal gets followed depends not on the amplitude (unless the amplitude difference is big enough), but on (1) the music (which is doing transient drifts to the carrier phase) and (2) the transmitter phase drift, and (3) the receiver (which will have a drift preference at any moment).
It is possible to make FM receivers which are not phase-locked. You can, for example, convert the frequency into amplitude before doing conversion. In that case, your mixed FM signal wouldn't work at all, because the noise wouldn't have a recoverable carrier frequency.