Are there "interesting" (that is non-trivial, for example not containing only one set) set theories with one element set being equal to their element ($\{x\}=x$ for every $x$)?
This question arose from the practical problem: Is it possible without "troubles" (such as contradictions) to consider an RDF term denoting a transformation between XML namespaces as an one-element set containing this term? If we can consider them equal, it makes shorter the notation, as we do not need to define an one-element set in this case but use the transformation term itself to denote this set. To define one term less in this case.
Yes. If you take out one axiom (the Axiom of Regularity, which roughly disallows sets to be nested inside themselves), then ZFC is perfectly happy with the existence of sets $x$ such that $x = \{x\}$. Such sets $x$ are usually known as Quine atoms.
In fact there are many well-known set theories that explicitly allow the existence of Quine atoms, sometimes as a matter of principle --- New Foundations, for instance.
Of course it will never be the case for all $x$ that $x = \{x\}$. This would give a contradiction. Specifically, we can prove that $\varnothing \ne \{\varnothing\}$. If you wanted to change this, you would have to change the very definition of membership or equality of sets.
Given your intended computer programming application, I will also mention that there are programming languages which treat $x$ and $\{x\}$ as the same thing. This is not a problem, there is no contradiction, because "sets" in these languages are much more restricted than the sets of set theory. Typically, $\{\varnothing\}$ will not be an allowed set, and in general there is only a single level of nesting (i.e., no sets-of-sets).