Q:

Do we fly full size aircraft with video game style joysticks?

Are there planes with electronic joysticks as shown below?

RC airplanes are incredibly effective at using this technique. For actual planes it would remove the need for rudder pedals and a stick along with potentially being easier to get used to.
Are thumb controlled sticks used in aviation? or is there a reason to why they aren't used?

A:

They are used, they are just bigger joy sticks than used for RC flying.  
A current example is the Airbus A320 (and related family) of passenger aircraft.  A short video of the joystick in action(called "side stick", start at about 1 minute in).  There is no direct mechanical linkage between the stick and the flight controls. 

(For the A330, but basically the same). 
For more detail, you can download this (slightly out of date, but with enough information to be useful) Flight Controls lesson for the A320 here. 

I am more curious about thumb controlled flying as you could control
  everything from what essentially looks like a video game controller.

@Rory Alsop noted that ... 

... even video game controllers are terrible for this. The bigger the stick the better fine control you have over it. This is why for flight
  sims for PC's and consoles, keen players buy proper joysticks -
  otherwise they are twitchy and hard to handle

Video game controllers require two hands, have poor sensory feedback (as compared to a stick type controller) and are not as ergonomically effective as a multi-axis stick.  Also, there are some "push buttons" on a control stick that allow moving a finger or a thumb to activate other functions, like the "take control" button in the side stick(For swapping control between left seat and right seat pilots).  On some sticks you'll also have the "push to talk button" for the radio, as has been done for decades.  It's an efficient use of the hand, without having to move the hand for multiple tasks. 

A:

A thumb operated joystick requires fine motor skills while a big lever on the same joystick (essentially the modern control device as answered by KorvinStarmast) provides the same functionality but with larger movements required to make them. This is helpful for subtle control work and makes piloting in turbulence feasible, something RC flyers don't really have to contend with while standing on the ground.  

A:

As others already said, joysticks are used. But there is a fundamental difference with the RC controls.
We humans do much better job of dynamic control when controlling the force rather than position/displacement. It is too long a discussion why, but the main reason is that we have special force feedback sensors in our muscles. (Yes, it's a common misconception that we have only five senses).
For this reason, airplanes are generally controlled by force. Even those that have mechanical links and deflect the controls proportionally to position; even those FBW that measure position of the controls (the first type that Peter Kämpf mentioned) - they still, in the end, are controlled by human force. This is why correct loading (spring or otherwise) is crucial: certain displacement requires certain muscle force, and this force we apply and control.
Ideally, you want to tune the loading so that a certain force would produce a certain desired end effect. Say, each 1 kg pull produces additional 0.1g of normal acceleration. This naturally happens with classic reversible controls: when airspeed increases, the required control deflection (for a given acceleration) decreases significantly. But the required force per unit of deflection increases by the same amount. Thus we have constant force per acceleration, despite having very different control travel. This travel difference can be an order of magnitude, but it poses no problem for the pilot, because force is the same. In fact, pilots use it as a secondary feedback about airspeed.
Electronic (FBW, fly-by-wire) joysticks could be made to require no or little force, like RC controls, but this is never done. A great deal of effort is invested in producing optimal loading. For one, an accidental move of a light control might destroy the airplane. (This may happen even with RC airplanes, which are much stronger relatively). But even more importantly, it has been proven that gradual precise force control works better for us that position control.
This is why some aircraft (starting from F-16) moved away from positioning entirely and measure force on the stick directly. However, this just doesn't feel natural for us: we expect some movement in response to a force; so a small amount of movement is generally allowed even for pure force control.
The amount of travel also matters, but this is only a secondary feedback. Besides, it works better when we see the movement, which doesn't always happen when flying.
When you press brake in your car, you also feel and coordinate mainly the force on the pedal rather than its position. This allows you to adapt to different cars quickly. Racing cars have very little pedal travel. But if the required brake force was very light, you wouldn't be able to brake smoothly, no matter what travel there was.
Now the RC controls have almost insignificant spring, and are actually controlled by position. Normally, this doesn't give a precision anywhere near what a real airplane would require. As Daniel noted in his comment, you wouldn't be a happy passenger on such an airplane. This is partly mitigated by using a very close position reference (the hands grab the sides of the control) and using our most sensitive fingers, but still only partly.

