Why Force is called a vector quantity? We have quantities which just adds up simply irrespective of any order and we call them scalars. 
Then we have quantities which adds up algebraically according to their relative directions and we call them vectors. Under special conditions they adds up according to scalars.
So, given that Force adds up according to its absolute direction that is we additionally have to consider the line of action here just as we additionally have to consider the (relative) direction of quantities in vectors, shouldn't it be something more than a vector and classified accordingly?
Also, what are the other such type of quantities?

PS- Should I write Torque instead of Force since most of the time we don't have to account for the line of action of Force. If this is the case then feel free to edit the question.
 A: A single force can be represented by an arrow (a simple assumption)

An arrow is one-dimensional. 
As by Google, a 
Vector is:

a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.

Quantity is:

the amount or number of a material or abstract thing not usually estimated by spatial measurement.

We know that the length of an arrow represents the power of a force, or in other words, the quantity. Also, we know that the direction of an arrow is it's, well direction, and the magnitude being the quantity. And, also, it is actually determining the position of a point in space relative to another! The comparison is, well an origin on a set of coordinates, longitude and latitude in real life, and there are also many more examples! 
From this, we conclude that it is not necessary to classify as something else. 
As for your last question, I don't understand what is meant by such quantities, so therefore, I cannot answer that (probably because of my poor comprehension...) 
A: In the experiment, the two pushes have the same force, but different torque. In that kind of analysis, the forces are free vectors, so is, equivalence classes of oriented segments: each vector is the class containing all segments with the same length, direction and sense. In the diagrams aren't the vectors drawn, but a representative of the relevant class. In the analysis of the pushes we write for the first:
$\vec\tau_1=\vec r\times\vec F$, measured respect to some point in the chest. The second is $\vec\tau_2=(-\vec r)\times\vec F$ same symbol because the force is the same.
So, you can push with the same force both shoulders and no torque because $\vec\tau_1+\vec\tau_2=\vec r\times\vec F+(-\vec r)\times\vec F=(\vec r-\vec r)\times\vec F=0$ (and $M\vec a=\vec F+\vec F=2\vec F$, too), So is, you need identify somehow the, apparently, different forces and it's done by means the concept of equivalence classes.
The two different pushes seem in fact different and we first think of the forces, but the theory says where is actually the difference and it's not in the forces.
