The phenomena of surface tension is very confusing!! Why it is a force per unit length not a force per unit area like pressure?
It is the force which a film of length L can sustain. Hence surface tension is defined in this manner. However, if the force acting perpendicular to an area is considered, it is pressure.
In addition to what has been said, it also makes sense to understand surface tension as the energy stored in a surface per area. The total surface energy of a system is then
E=total area*surface tension
where the equivalent unit of surface tension is J/m^2 (compared to your N/m).
In order to compute the actual force it is not sufficient to just know the surface tension (which is the material parameter of the interface or surface) but you also need to know the "geometry" of the interface. For example, the Young-Laplace equation additionally requires the mean curvature. Then the force follows from considering variation of the surface energy "E" with respect to changes in the surface (shape and size). This is also true in a rigorous mathematical sense. Hope that helps.
Actually we can think of surface tension as being analogous to tension in a stretched membrane. As that tension is measured force per unit length of the membrane, (that length will be perpendicular to the force) it makes more sense for surface tension to be taken per unit length.
Secondly, the definition makes more sense to explain the phenomena like needle on water as needle essentially has one dimension.
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