I believe multi-mode fiber does carry more than binary data -- though I think its split into different channels, rather than one big channel. Regardless, channels can always be teamed together to have the same effective bandwidth increase.
The reason its not done the way you describe is that its always easier to distinguish simple on/off binary signals, than to measure and categorize signals with multiple levels. It's also a more-robust signal. As an analogy, imagine you are trying to communicate non-verbally with your friend across a dark room. You've decided to exchange messages by using ASCII codes. To signal a zero, you hold up a black card and to signal a one you hold up a white card. But, you say, "Aha! this would go much quicker, if we signaled two bits at a time by using four cards -- black, dark grey, light grey, and white!" Mathematically, this is sound, but it also increases the chance of error -- maybe, depending on the ambient light, you sometimes confuse black with dark grey, dark grey with light grey, and light grey with white. Maybe you can slow down to be more sure of your decoding, but then you've defeated the purpose of using multiple levels. The problem gets worse the more levels you add to your signal, and all of this is to say nothing of attenuation or a weakening signal.
At the same time, another means of increasing total bandwidth is simply to run the binary signal twice as fast. Assuming you can keep up, this method is preferable because there's no need for the error-prone and slow process of decoding various signal strengths. For various reasons and properties of electromagnetic signals (including light), I'm reasonably sure that it always holds that a faster binary signal is more practical than a slower multi-bit one -- at least in so far as transmitting discrete data (that is, information that is not naturally analog) is concerned.
There's an analog to be made to Solid-state storage (like SSD hard drives or SD cards) in that the fastest, most-reliable storage uses Single-Level-Cell memory, which stores just one bit per NAND cell. Because this is expensive, its mostly limited to enterprise and enthusiast SSD drives. Common high-performance drives today use Multi-Level-Cell memory that stores two bits per cell in a manner similar to what you describe (using different voltages to represent different bit patterns), however, these cells are slower to read and to write, and are less stable than SLC cells. Because MLC is *still* relatively expensive, we're now starting to see SSD drives with Triple-Level Cell memory, which is even slower and even less stable than MLC, but is around half to 2/3rds the cost per gigabyte. For flash memory chips, MLC seems to be the commercial sweet-spot at the moment, but you're dealing with different design parameters than fiber-optics.