What exactly is a variable in C++?

Variables are names that you give to objects, so yes, objects are, by and large, anonymous.

Variables are named objects. The following create objects that are not variables

new int // create one int object
std::string() // create one string object

The following creates one array variable with name "foo" and 5 unnamed (sub-) objects of type "int"

int foo[5];

The following is not a variable in C++03, but has become a variable in C++0x (declared references are variables in C++0x, for details see the link)

extern int &r;

---

Does a variable give a name to an object, i.e. are variables just a naming mechanism for otherwise anonymous objects?

Variables are objects (or references respectively). The entity list (3/3 in C++03) of C++ contains multiple such is-a relationships. For instance, a sub-object is-a object and an array element is-a object and a class-member is-a object or function or type or template or enumerator.

The entity list of C++0x looks a bit cleaner to me, and it doesn't contain "variables", "instance of a function" (what that kind of entity even is has never been apparent to me), "sub-object" and "array element" anymore. Instead it added "template specialization" which either are functions, classes or templates (partial specializations).

The C++ object model at 1.8 says

An object can have a name (clause 3).

So if you like, you can formulate the statement as "The object's name denotes the object.".