There was a thread on this a few months back ... yes, we (ATC) can declare an emergency for a pilot. There are many reasons but the most common would be for our own piece of mind, during an emergency a pilot (or flight crew) is very busy and we (ATC) only know a fraction of what is going on in the cockpit. There is no penalty in declaring.
When a pilot states he is having difficulty we do everything we can to provide assistance. In the case of declaring an emergency it gives us more tools to work with. On any given day I would prefer to overreact and declare an emergency then to look back and wish I had.
Many GA pilots (and this is not a slam against them) seem hesitant to declare and emergency. I would speculate this is caused by fear of answering to the FAA, or not being familiar with ATC (dealing with grumpy controllers ...LOL). There are any number of reasons, but I have declared for GA A/C. I sensed hesitation in their comms, hard to explain, but in one instance I spoke with the pilot later in the day and he thanked me, said it helped him to just fly the plane.