Is it possible to make a beverage using sugarcane juice or cane syrup? Not by distillation (rum), but like beer or wine? There’s tons of this here in Florida.
its possible to ferment anything that has sugar in it. Does it taste good is probably a better question.
Fermenting it would be the first step to making distilled spirits like rum, so yes it is possible.
As I understand it, mead makers have to add a lot of yeast nutrient because honey is rather deficient in that respect. Probably the same would apply to sugar cane juice. If you want to try, I suggest looking into the yeast feeding procedures for mead making. It may well taste terrible, or take months of aging to become drinkable.
Sure, “sugar wine” or “Kilju” or a form of “prison brew” … although those are usually made with refined sugars … and they’re not known for their drinking quality. 
However, making sugar wine then adding fresh lemon juice in the secondary can produce a drinkable lemon wine. Getting sugar water to ferment clean, especially if the gravity is relatively high, requires a LOT of attention to details: careful management of nutrients, pH and temperature … and stepped additions of the sugars.
I believe unrefined cane juice will have at least some of the required nutrients since some craft distilleries use panela (basically dehydrated raw cane juice) for their rum … without using nutrient additions. But rum and sugar wine are two totally different animals.
If you just chuck yeast into sugar water and hope for the best, like the Finnish punks do with their Kilju, don’t hold your breath waiting for compliments.
It’s usually necessary to make a wort with malted barley and then thin it with cane juice or syrup (10%).
Mostly these are fermented cold with lager yeast.
Fermenting only cane juice/syrup without distillation normally results in the production of higher alcohols, especially without temperature control and because of the highly fermentable nature of the sugar itself which leaves very little residual flavor behind.
I think you could just treat this like a session-style mead. Make a base brew in the 8-10% ABV range (using the same good fermentation practice you would for a mead - staggered yeast nutrient, temp control, degassing, etc.), then hit it with sulfite/sorbate to stop fermentation. Then backsweeten with more cane juice. I’ve thought of trying this myself with using a base mead aged on some cane with lime juice additions - sort of a caipirinha mead.
