It's a weird quirk that is unique to the upright bass, but "Full Size Bass" is a bit of a misnomer. Every other stringed instrument (and a selection of other instruments, as well) consider 4/4 size to be the full-size, common instruments for adult players. Most have "downsized" (aka "fractional") instruments that are generally used for younger students, with their smaller limbs, and players generally are expected to graduate to the full-size instrument when their hands and arms grow larger. But in the bass world, we're a little different; the commonly recognized "regular-sized" bass is actually the 3/4 size bass. But yet, a 4/4 "full" sized bass still exists. However, most bassists/teachers/stores (including me) will usually tell you that you don't want one of them, except in fairly rare circumstances*. Continue reading to find out why!
So I've done a lot of reading on the subject, and while I'm quite far from any kind of an expert on bass history, I strongly believe that it's a simple practical matter of the evolution of the instrument. Because the bass - and more pertinently, its strings - have improved over time, it made such a physically huge instrument less necessary for volume and tone/pitch quality.
Originally, the instrument was called a "double bass" because it was created for use in very large orchestras to "double" the cello parts, one octave down. This gave a newly grand, full-bodied, and powerful low end to orchestras of the time. The double-basses used all-gut strings (which was all they had at the time), and were, therefore, necessarily huge, unwieldy instruments. The basses (and strings) had to be large, in order to give any volume or tone to those very low pitches. As you can rightly imagine, with an imposingly large scale length and massively thick, floppy strings, they must have been an absolute beast to play.
The development of the over-wound gut string, and later, strings made of steel and other materials -- greatly improving the tone, pitch, and playability of the ere-while massively thick low strings -- quite likely saved the laborious double bass from extinction. This improvement in strings alone provided access to much better-sounding lower notes, while also allowing the instrument to be resized to its far more practical current dimensions, which was dubbed a "3/4 size" bass, since it was smaller than the previously existing basses.
Being easier to play and move (and sounding just as good for almost all circumstances), most players naturally gravitated toward the slightly smaller 3/4 bass and never looked back. And as such, it pretty much became the de-facto "standard" upright bass size for adult players.
While the size nomenclature can be confusing, especially to those new to the instrument, it all helped to evolve the bass into a legitimate instrument of its own standing, no longer relegated to simply doubling the parts of other instruments.
You'll find that the 4/4 size bass is really considered more of a "jumbo" or "XXL" bass - and mostly they are used by bassists in large orchestras, where maximum acoustic volume and fullness is very important -- as a relatively small bass section has to carry the low tones for the string section acoustically. Additionally, the extra depth can be advantageous for those using E-string extensions for added low notes (usually down to a low C), again, for maximizing the particularly low tones produced by the extra low string.