Firstly, I don't care what a cider is OFFICIALLY defined as. If its bubbly, made from fruit, and not strong enough to call a sparkling wine, I'm calling it a cider for lack of a better term. And nobody can stop me, because I homebrew so I get to put whatever I want on the label
NOW to actually respond to the OP:
I wondered the same thing, and made a gallon of cider with Pineapple Orange Juice awhile back. I wasn't really sure how to account for differences in acidity or sugar content, so I just put in a little more yeast nutrient than I would've for apple juice.
I fermented the straight juice (no sugar added) on a US-04 yeast cake from an apple cider that I had just bottled, thinking that the existing cake might have a better chance surviving the acidic conditions than a new packet trying to get started. It fermented like crazy, so that worked.
I transferred to bottling bucket after two weeks, and it smelled like VOMIT. I thought it might be ruined, it was very bitter and tart, so I backsweetend with 1 can apple juice concentrate. Still tasted too tart, so I also added a can of Old Orchard 100% berry blend Concentrate to hopefully salvage something sweet enough to be drinkable. SWMBO thought it was ok after that.
Ironically it fermented to 6.66% ABV, and tasted like death, so I quickly labeled it Super Devil Juice on bottling day, and chalked it up as a failed experiment. I was pleasantly surprised when I tried it carbed, pasterurized and chilled. SWMBO liked it as well. Also, the 2 cans of Concentrate I backsweetened with to "save" it caused it to carb up quick before I pasteurized, so it foams like a banshee when you crack a bottle, but its drinkable.
I think with citrus, moreso than apple, the carbing and chilling makes a big difference. Kinda like the difference btw warm Dr. Pepper and ice cold one. The acidic, pukey taste is subsided, and more of the (other) fruit flavors balance out some of the citrus bite after it had some time to age and chill.
The Berry blend kinda carries the flavor (i was hoping it did), but the citrus is still very present, and you cant even taste the apple. Im not sure how this wouldve tasted if I had stuck w citrus and backsweetened with a can of Orange Juice Concentrate, but even with the delicious Berry Blend's help, its not an experiment i plan to repeat.
For the finished flavor I got, I wouldve rather started with a Berry blend juice (which probably has apple juice as first ingredient anyways), and backsweeten with the BB. Then Id get the berry sweetness without the citrusy tartness, which I think wouldve been addition by subtraction.
I think the pineapple really what made it gross, as they can have that insipid taste sometimes, so this might have better success as just an OJ beverage without pineapple. Unfortunately, I bailed out at bottling time in this citrus journey, so if someone else is brave enough to try this, try to ignore the smell when backsweetening and just realize it will taste sweeter when its cold and bubbly. Maybe using sweet citrus fruits like clementines or mandarin orange would also yield better results. Just say no to Pineapple, and I wouldn't try grapefruit either.
Good luck, hope that helps OP.