The plug in the picture has plastic "wings", which seem to have no other purpose than to irritate when trying to insert multiple devices into a power board. The wings do not allow cords to be plugged in on either side of the plug.
I am wondering why they are there. There is nothing inside of them and no damage at all would occur if they were removed. This particular example is for an external hard drive, but I have seen several plugs of this sort for various devices.
This is an Australian plug. The only other information I can find on the web is this old forum post which does not give a clear answer.
Best Answer
Those 'wings' are grips. In the pictured module, the contacts are mounted in a removable 'faceplate.'
In this case, it's an AUS-compatible faceplate, and those wings are to ease installation & removal of that plate.
More to the (assumed) second intent of your post: Yes, you should be safely able to remove them safely.