Yes hot chocolate does have the ability to stain your teeth.
The hot chocolate contains chromogens which can discolor your teeth and stain them if you do not brush your teeth soon after drinking the hot chocolate.
Chocolate milk and any other chocolate can also stain teeth.
But if you brush your teeth at least a few times per day which you should anyway then you should not have to worry about your teeth staining.
I've never had my teeth stain from chocolate, hot chocolate etc because I brush my teeth daily and also at least once per month I use teeth whitening strips.
But some people have gotten stained teeth from chocolate as well as hot chocolate.