Q:

what is the arguments variable an instance of?

MDN says it is "an Array like object" but does not say what it is an instance of.
It is not an HTMLCollection or NodeList.
If I call Object.prototype.toString.call(arguments) it returns "[object Arguments]" but arguments instanceof Arguments is an error.
So what is arguments an instance of?

A:

So what is arguments an instance of?

It's an instance of Object.  There does not appear to be any public Arguments constructor that you can use with instanceof to identify it that way.  
If you want to uniquely identify it, then:
Object.prototype.toString.call(arguments) === "[object Arguments]"

is a safe way to identify it.
Per section 9.4.4 in the EcmaScript 6 specification, the arguments object is either an ordinary object or an exotic object.  Here's what the spec says:

Most ECMAScript functions make an arguments objects available to their
  code. Depending upon the characteristics of the function definition,
  its argument object is either an ordinary object or an arguments
  exotic object. 
An arguments exotic
  object is an exotic object whose array index properties map to the
  formal parameters bindings of an invocation of its associated
  ECMAScript function. 
Arguments exotic objects have the same internal
  slots as ordinary objects. They also have a [[ParameterMap]] internal
  slot. Ordinary arguments objects also have a [[ParameterMap]] internal
  slot whose value is always undefined. For ordinary argument objects
  the [[ParameterMap]] internal slot is only used by
  Object.prototype.toString (19.1.3.6) to identify them as such.

Since it is an "exotic" object, that essentially means it doesn't follow all the normal and expected conventions.  For example, there is no constructor function from which you can create your own object.  And, because there is no public constructor function, that probably also explains why there's no instanceof that you can test on it to uniquely identify it.

