This property ( or rather the generalized version by Darij using (a,b)-shuffles ) means that f is what is called a "symmetral mould" in the context of Ecalle's theory of moulds. There is a related notion of "alternal mould" where the right hand side is 0 rather than a product of two f.
Here is just one reference among many : page 591 of
This may not be transparent when looking at this article. Maybe page 2 of my article
would be more clear, but it only defines "alternal moulds".
The symmetral property is really a property of sequence of functions $f_n$, with $f_n$ a function of $n$ variables $x_1,\dots,x_n$.
The notions of alternal and symmetral moulds, when considered under some specific point of view, turn into the notion of primitive and group-like element in a Hopf algebra.