Why does AFSK only trasmit in binary?
The three main reasons appear to be these: 

1) In AFSK, it's not just jumping back and forth between frequencies. The tones must also be continuous phase. In other words, when the tone changes, there can't be any jump in phase. For example, if you're sending a 1200 hz tone and the waveform is at its peak when you switch to 2200 hz, the waveform is still at its peak, so it can't be just any two frequencies. It has to be at the peak. It can't start back at zero, or halfway through; it can't start anywhere but at the peak. Additionally, the length of each tone must be the same. Because of these two points, the amount of wavelengths that share compatible periods is finite. For instance, if you use 1000 hz, your next compatible is something like 1500 hz (my apologies if my numbers are off; I'm worn out).

2) As a follow-on from 1, since AFSK is audio, the range of compatible frequencies is not that high.

3) The amount of bandwidth required for the transmission increases for each additional tone, making data efficiency greater, but power and bandwidth efficiency less.

That doesn't meant that multiple frequencies can't be used; in fact MFSK has been used since the mid-60s. However, MFSK has several drawbacks that make it untenable in some situations. Delay spread, when broadcasting to multiple locations, and fading make it less desirable for some applications.