What does the emergence of different interpretations of Wittgenstein signify?
I'm going to answer the question from a different angle: the competition of interpretations have nothing to do with Wittgenstein's work per se, though his work may illuminate why multiple interpretations occur.
In the broadest sense, it signifies properties of natural language: ambiguity and vagueness. But, I'd add a third significant factor: first principles. That Wittgenstein, particularly through his notions of family resemblance, has shaped the conversation that resulted in the linguistic turn or, say, the nature of definition (particularly through prototype theory), in modern linguistics is coincident.
Philosophical arguments come in roughly two flavors: one, where everyone agrees on the general ideas that go into a posteriori reasoning, and two, where there's much ado made about a priori reasoning that precedes discussion. In linguistics, one might hear the phrase 'semantic frame'. In rhetoric, this is observed in the notion of 'framing a debate'. In law, one often hears about 'arguing the facts'. Most broadly, sociologists call this (unsurprisingly) framing. From WP:
In the social sciences, framing comprises a set of concepts and theoretical perspectives on how individuals, groups, and societies organize, perceive, and communicate about reality.
In the language of logic and argumentation theory, one could say there are arguments about premises, and there are arguments about inference. And in fact, to make things more confusing, since there is a body of theory about inference itself, one can argue about premises related to inferences, and about inferences about inference. The end result of this is that philosophical discourse which is conducted through natural language is inherently open to interpretation, an idea that snaked its way right into modern philosophy in ways such indeterminancy of translation and underdetermination of theory being two.
Can the emergence of different interpretations of Wittgenstein be seen as a failure of his thesis?
Not at all. Actually it's a confirmation. You invoke the notion of language games, and to distill, this is just the notion that agents who use language derive and invent meaning based on a normative or goal-oriented basis. That is, it is important to see language as an exchange of sound with the agendas of the agents involved, a notion that is in essence an organizing principle of pragmatics. And if different agents (read philosophers) have different values and goals, shouldn't we expect that they attempt to create and use language to achieve different ends, and thus they differ in their artifacts such as definitions and theories? Of course!
Furthermore, this is important to understand, because it leads into a very revolutionary idea at the time, the notion that the meaning of language, semantics, is not strictly speaking a property of an entity called language, but a process of interpretation of agency. This is part of the focus of the linguistic turn, where the philosophy of language experienced some serious growth.
Of course, this wouldn't surprise LW at all. The preface to Tractatus starts thusly:
Perhaps this book will be understood only by someone who has himself already had the thoughts that are expressed in it -- or at least similar thoughts.
So, what's up with the interpretations of LW's work? Maybe the best starting point to answer that question is what's up with the interpretations of Aristotle's? Plato's? Kant's? Russell's? What's up with the nature of language and interpretation to begin with?
