Would it be feasible to make a chocolate-flavored cheese? The idea that originally occurred to me was the use of chocolate milk for this purpose, but I'm not sure that's the best solution. Does this food already exist? What kinds of cheese are doable in this situation?
The other answers refer to products that are cheese-flavored fudges or cream cheeses blending with cocoa.
Your question seems like it is asking about making a cocoa flavored cheese from scratch.
I have not tried chocolate in particular but I have been experimenting with flavoring cheese (and tofu) and have some data points that might be helpful.
I have not had much success with flavoring the milk before it is curdled. The curdling and pressing process is designed to force out water and water soluble compounds. I would expect much of the cocoa dissolved in the milk to be wasted. Additionally, I wouldn't expect store-bought chocolate milk to work very well because of all the additional sweeteners, emulsifiers, and thickeners that are included. I don't know how it would behave when making cheese but I wouldn't be surprised if they got in the way of forming a good curd.
I have had more success with adding dry flavorings to the curd after it is drained and before it is pressed into the final shape. This is the same time that the salt is added. Some water is still pressed out and so some flavor is lost but much less than in earlier stages.
I am still experimenting with how much dry flavoring can be added before the curd doesn't set properly. The cheese becomes more fragile the more ingredients interfere with the milk proteins.
An easy way to start would be with a simple paneer or queso fresco. Simply heat up milk, stir in some acid (lemon juice, vinegar, citric acid, etc), let the milk protein precipitate out, strain it, mix in salt and some cocoa powder, and let the mass drain in a cloth until it is the consistency that you want.
When thinking of food combinations, I try to think similar combinations that exist already.
I'm sure I've had cheesecake with chocolate decorations. Without a doubt, the flavor combination works in that type of food. Likewise, I've also had chocolate cupcakes with cream cheese frosting, as well as chocolate cake-pops with cream cheese frosting.
If I had to generalize, I would say keeping the two flavors discrete (i.e. frosting has one, cake batter has another) seems to have better results. I haven't heard of chocolate going with heavily salted (or otherwise flavored) cheeses, but this isn't to say that such a combination can't be done well.
It's OK, if you don't think of it as cheese.
I never actually tried it, but at Michigan State's Dairy store, they sell chocolate cheese.
This fudge-like delicacy blends the finest cheese with sugar, vegetable oil, peanuts, and chocolate powder.