Given that some MMA users would insist that one should not use the '$' sign in function names anyway, this is not reported here as a bug, but it is frustrating given also just how useful the '$' sign can be.
For most valid MMA function names (function expressions) like testFindMe[x] if it is used in, say, one .wl package, CMD+clicking on it will navigate to a window with the .wl with the private implementation (as opposed to the ::usage). If the .wl with the private implementation is not already opened, it opens it. (It does not seem to be smart enough to navigate to implementations with different argument patterns like testFindMe[x,y].)
has$two$dollar[x] breaks this function search system. You can in fact see how it breaks as it highlights part of the function name separated by the '$' sign.
It's not clear whether it's a limitation intrinsic to Eclipse IDE, or due to the design of Wolfram Workbench.
Feature suggestion: It would also be very nice if clicking on the (`Private`) implementation of a function took you back to the ::usage.