Holographic principle and string theory are related in some way or completely independent? Are the Holographic principle and string theory related in some way or are completely independent?

  
*
  
*Everything is happening in a surface and our three-dimensionality is
  an illusion of our senses.
  
*On the other side, string theory assume the existence of 11
  dimensions.

 A: As stated in another answer already, your definition of the holographic principle is misleading. More precisely, the holographic principle says the following: A theory of quantum gravity in a compact space should be equivalent to some theory on the boundary of this space. This statement (which is a conjecture) is motivated from the result of Bekenstein and Hawking, that the entropy of a black hole is proportional to its boundary area and not to its volume. This is a priori completely independent of what this theory of quantum gravity is.
One particular proposal for a theory of quantum gravity is string theory and there are indeed hints that this theory is holographic. These hints come from the AdS/CFT correspondence or, more generally, gauge/gravity dualities. The AdS/CFT correspondence is a well-tested (but not proven) equivalence of type IIB string theory on asymptotically AdS backgrounds and a quantum field theory on the boundary of space. This gives us some hints that string theories do indeed obey the holographic principle. However, the AdS/CFT correspondence only makes a statement for particular solutions and is itself not proven. It is particularly nice though, that the boundary degrees of freedom are organized as a gauge theory. From the point of view of the holographic principle, the boundary theory could be anything (and therefore, it could be much more exotic).
Aside: Your statement that string theory asumes 11 dimensions is not correct. The spacetime dimensionality is a prediction of string theory and is derived from other principles. It is not an assumption.
A: It is conjectured that they are equivalent. Regarding your two remarks notice that:
1) In string theory the extra dimensions are curled and small enough that we cannot perceive them. So for any practical everyday purposed string theory results in a 3D world.
2) the quote:

Everything is happening in a surface and our three-dimensionality is an illusion of our senses. 

is often misleading in popular science accounts. If two theories are equivalent there is no way to tell scientifically wich one is the correct one. That belongs to metaphysics. 
