Why can we factorize the state of a particle? I read about factorization in these two cases:

*

*When spin and position are not coupled it is
possible to factorize the state in a wave function and a spinor $|\psi\rangle|\chi\rangle$

*If there are two independent particles, it is possible to write the wave function as a product of wave functions  $\psi(\vec x_a) \psi(\vec x_b)$
I don't understand the logic behind it, is it a math property or a quantum axiom?
 A: Yes, the factorizability  is an axiom. Formally, it is states as 

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems (for instance, J. M. Jauch, Foundations of quantum mechanics, section 11.7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles. 1

It can be essentially paraphrased as 
When we want to combine degrees of freedom we take their tensor product $\otimes$ of their corresponding spaces.
Why? It comes down to the Linear Algebra formalism. Suppose a hamiltonian looks like 
$$H = H_{spatial} + H_{spin}$$
Well... kind of. This way of writing it is mathematically imprecise. Really the hamiltonian $H$ lives in the hilbert space $\mathcal{H}_{pos} \otimes \mathcal{H}_{spin}$ (since we are combining degrees of freedom), and so the correct way of writing it is 
$$ H = H_{spatial} \otimes \mathbf{1}_{spin} + \mathbf{1}_{pos} \otimes H_{spin} $$
which jives with the above postulate.

Edit: As Dan Yand pointed out in the comments, it is important to note that my answer refers to the axiom of the factorizability of the Hilbert spaces, not to the factorizability of a particular state. In general one cannot always do the latter.
A: This is not an axiom but a mathematical property. In general a two particle state is the function of the two particles. Only in the special case that the two particles are uncorrelated, such a function can be written as a product of two one-particle wave functions. Multi-particle states of identical quantum particles can in general not be written as products. For example, many-electron states have to be written as (expansions of) Slater determinant functions.
As to factorization into a spatial and a spin orbital, this is possible only if the spin is not correlated to the spatial coordinates.
