NP-hard: an instance or a set?
You seem to be confused about terminology. You start correctly: "a problem is NP-hard". This already answers your question: hardness is a property of problems, not sets of problems or of instances.

To refresh, a problem is a question of the form, "Does the input object have this interesting property?" An instance is a specific object you're asking about.  So, for example, 3-colourability is the problem, "Can the vertices of the input graph be coloured red, green and blue such that there is no edge between vertices of the same colour?"  When you ask that question of a specific graph $G$, then $G$ is the instance.  (And a set of problems is just a set of problems: you know what a set is and you know what a problem is.)