What caused my homemade "Nutella" to go bad?

This is a basic fact of food safety. It doesn't matter how long each of the ingredients take to go bad separately. Prepared food will go bad soon unless you do something special to preserve it.

In your case, you had hazelnuts, which don't go bad because 1) they have too little water, and 2) bacteria cannot enter their tissue, which is made of intact cell walls. You also had chocolate, which doesn't go bad because it doesn't have enough water for bacteria to survive in it. Ditto for sugar. Then you mixed everything together and added water, the one thing which was missing in order to keep bacteria out. And you also pureed it, which destroyed the protective cell walls. Bacteria found your food and started multiplying in it.

To address your comment too: water by itself doesn't usually go bad, because it has all the water a bacteria needs, but not nearly enough food content. So, bacteria cannot survive in it, at least for a short time. Were you to leave out a mix of water and carbohydrates out, you'd notice vigorous bacterial activity - this is what fermentation is. But there are some classes of bacteria which can survive on the minimal amount of debris caught by a water reservoire sitting around for weeks, and one of them is indeed pathogenic, causing a dangerous form of pneumonia. So, water does deteriorate, just not as quickly as the usual types of wet food, which are hospitable to many more pathogens.

In general, all human food is great food for bacteria, often including pathogenic bacteria. You have to take something away to make it inedible for bacteria. Any mixture of two shelf stable foods can turn out to add exactly what the foods separately lacked, and make it a bacterial breeding ground again. So, never assume anything about food shelf life based on the shelf life of its ingredients!