What is friction? The component of the contact force parallel to the surface of contact is defined as the frictional force. But looking at the microscopic origin of the force I think it need not to be so.
Microscopically the attractive force between any two nearby atoms in contact is friction (this occurs due to the formation of cold bonds which leads to the stability of the system) whereas the repulsive force is the normal reaction.
Considering the case of a block lying on a rough surface and let's assume that the planet on which it's resting isn't performing any accelerating motion. Now clearly cold bonds would have formed (For simplicity assume that planet and the object are made up of pure metal without any coating of any type). Now when I try to lift the object the force which I would require would clearly be greater than ($\gt$) $mg$. This means that here the frictional force was applied perpendicular to the surface of contact (if not then why not?) and hence the given definition of friction isn't valid over here. So:


*

*Can we say that in lifting case frictional force is acting?

*Why is friction defined this way? 

*Is there any better definition of friction? 
 A: Friction is a force that acts parallel to the interface of two contacting surfaces opposing relative parallel motion between the surfaces. Forces acting normal (perpendicular) to a surface would not usually be considered friction forces. 
It should be noted, however, that not all forces acting parallel to the surface and preventing relative motion are the result of friction, i.e., the result of interlocking irregularities between the surfaces. Obviously the application of an adhesive would prevent sliding motion. Likewise, a cold weld would also prevent sliding. 
I believe in problems involving friction it is generally assumed that, in the absence of any compressive stresses, the contacting surfaces are free to separate in a direction perpendicular to the surfaces without the need to apply an external force.

That's why I assumed that metal thing. Also are you trying to mean
  that cold welding isn't friction?

When I researched it I didn’t see anything definitive. It is obviously due to attractive intermolecular forces that resist vertical separation as well as sliding. You asked if there is a better way to define friction. I’m suggesting that a friction force should be considered in the absence of any attraction forces. For example, Velcro does not hold things together by friction. It consists of an array of tiny hooks that Velcro is attached to. To pull then apart you must break the fibers  (pull apart the chemical bonds of he fibers.) 
Hope this helps
A: Frictional force is a contact force acting in opposite direction parallel to relative motion. If a force is perpendicular to the relative motion between surface, it cannot be a frictional force. If you stick two pieces of wood together, the force resisting its separation is not frictional force.
A: Placing a block like that on a wedge, the balance of forces is different from the typical ones.
Besides the vertical $P = mg$, the normal $N$, and friction $F_a$, there is also a $F$, resultant from the attraction of wedge and block surfaces. 
As $F$ is aligned to $N$, but has opposite direction, $N$ must be greater. It is now not only a reaction to $P$ but also to $F$.  $F_a = \mu N$ is also greater as a consequence. 
So, F is not a friction force, but increases it by increasing the normal.
