It depends on which part of your question you put the emphasis. If it is specifically the property of being anonymous for anonymous functions, then indeed the only answer is that they are unbound values. If you are talking about functions in general, anonymous functions are probably the most visible manifestation of the use of lambda calculus in a functional setting, for applicative languages.
In fact, from the lambda calculus point of view, lambda expressions are the very syntactic construct used to create bindings.
Recall the notation used in lambda calculus:
$\lambda f.\lambda x. f x$
where $f$ and $x$ are bound within the inner expression $f x$. Each lambda expression there defines the binding, and thus acts as a local context for name bindings.
A language usually offers ways, such as
let (ML like languages, scheme), or
define (scheme) to create bindings usable at the top level (or within syntactic constructs more complex than functions, like modules or objects), but the only needed tool for bindings is the lambda at lower levels.
If you look at languages like scheme or lisp dialects, their very fundation is lambda calculus, and many special forms are really sugar coated lambdas.
For concatenative languages, the story is slightly different. Lambdas are not necessary, and actually counter-productive. What's the point of defining anonymous lambdas, when everything is a function?
There is somehow a duality between these two kind of manguages. The later is focused on point free function combination and thus tries to represent everything as functions, while the former works on a more elaborated calculus, and makes an effort to have functions as first class values, like any other values of the language.
In this respect, one could see lambdas as the result of that effort.
Some pointers on the topic introduced (poorly) in this answer: