Well, well, well. Candy cane wine is one I have never heard of! However, as one of my winemaking professors always said, you can ferment just about anything as long as you can find the right microbe to do it! Candy canes certainly have a sugar source (sucrose), so wine yeast will definitely chew through that sugar and turn it into alcohol. Now whether or not you like the results is up to you . . .
However your question was about color loss. From my experience licking (and not fermenting) them, I seem to remember that most candy canes only have the red color on the outside. Lick the end into a sharp point (with which I loved to poke my sister) and what you’re licking is a white sugar core — you’ve sucked all that red color off. Most candy canes also are colored with food-grade artificial coloring. I’m no food additive expert but it’s no surprise to me that these colorings are not particularly stable. Because of their chemical composition under wine pH (acidity) and fermentation conditions it’s also quite possible these color compounds won’t stay red, or in solution, very long. Wine color compounds change all the time.
My suggestion if you’d like to maintain a little pink color in your candy cane wine is to add about 1–2 ounces of red grape juice concentrate per gallon of wine (7–17 g per L) you are making, before it ferments. This way you’ll introduce some more-stable color compounds into the brew but the “grapey” flavor won’t overpower what I imagine is a minty-fresh flavor profile you’re going for. Or, blend in some red wine before bottling, this gives more control over the desired color. I have to hand it to our creative home winemakers — candy cane wine is a new one for me! Good luck fermenting.