Why can vector components not be resolved by Laws of Vector Addition?

Indeed, any vector can be resolved in terms of two components (in $n$-dimensional space in terms of $n$ components). For this being possible the components should be linearly independent, i.e. in your case they should not be parallel. The advantage of using two orthogonal/perpendicular components is that their scalar product is zero, which simplifies the math when calculating the coefficients: \begin{array} \mathbf{F} = F_x\mathbf{e}_x + F_y\mathbf{e}_y \Longrightarrow F_x = \mathbf{e}_x\cdot \mathbf{F}, F_y = \mathbf{e}_y\cdot \mathbf{F}. \end{array} Indeed, \begin{equation} \mathbf{e}_x\cdot \mathbf{F} = \mathbf{e}_x (F_x\mathbf{e}_x + F_y\mathbf{e}_y) = F_x\mathbf{e}_x \cdot\mathbf{e}_x + F_y\mathbf{e}_x \cdot\mathbf{e}_y = F_x, \end{equation} and similarly for $F_y$, since \begin{equation} \mathbf{e}_x \cdot\mathbf{e}_x = 1, \mathbf{e}_y \cdot\mathbf{e}_y = 1, \mathbf{e}_x \cdot\mathbf{e}_y = 0. \end{equation} For non-orthogal vectors $\mathbf{e}_x \cdot\mathbf{e}_y \neq 0$, the math becomes a bit more complicated and the interpretation of the projections as coordinates is less intuitive.

There are however some cases where using non-orthogonal components is beneficial, notably when dealing with non-orthogonal crystal lattices, such that of graphene (a hexagonal lattice.)

Vadim's answer resolves the Q very nicely. I would just like to add that given a set of n vectors in a vector space you can always construct a set of n orthonormal vectors which can then be used as a basis. This is well known as the Gram–Schmidt orthogonalization process.

Suppose you are attempting to find a point on a map given a starting point. Which do you prefer:

  • 4 km north and 3 km east of your current location
  • 7 km north and 4.2 km southeast of your current location

We can use whatever non-parallel vectors we want to describe an offset in two dimensions. Is it now clear why north and east are preferable to north and southeast?