Does the Turing test, the attempt to verify the proposition "Machines can think" through an 'imitation game', come down to a confusion over "like" and "identical with"? i.e can I say the following "If it is like x is thinking, therefore what x is doing is identical to thinking"?

That's one interpretation, but there are many others. My favorite interpretation focuses on this passage in Turing's classic 1950 essay, "Computing Machinery and Intelligence" (Mind 59:433-460):

I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.

Of course, that century ended around the year 2000, and Turing's predicted "alteration" hasn't yet happened. But that's beside the point. Turing's claim, according to this passage, is that, if computers (better: computational cognitive agents or robots) pass Turing tests, then we will eventually change our beliefs about what it means to think (we will generalize the notion so that it applies to computational cognitive agents and robots as well as humans), and we will change the way we use words like 'think' (in much the same way that we have generalized what it means to fly or to be a computer, and have changed the way we use those words: Once upon a time, only animals like birds flew; now airplanes do, too. And once upon a time, only humans were computers; now machines are, too. (Take a look at an ad for a "computer" in The New York Times from 1892)

For more discussion, see:

James H. Moor (ed.), The Turing Test: The Elusive Standard of Artificial Intelligence (Dordrecht: Kluwer, 2003)

Shieber, Stuart M. (ed.) (2004), The Turing Test: Verbal Behavior as the Hallmark of Intelligence (Cambridge, MA: Harvard University Press).

Read another response by William Rapaport
Read another response about Mind