Is it an assumption or truth for spatial and temporal variables separation if Hamiltonian is time-independent in Schrodinger equation?

I'm not sure if this is what you're looking for, but (as AccidentalFourierTransform points out), the technique is called separation of variables.

The question is, how do we know all the possible solutions to the Schrodinger equation are of this form (separable) when the Hamiltonian is time-independent?

The way to check, which I understand is standard, is to look at all the possible general solutions you get out of the spatial and temporal ODEs post-separation, and see if their products $φ_i(x)T_j(t)$ form a basis over $L^2[(0,T)times mathbb{R}]$. This would mean you can match any function in $L^2[(0,T)times mathbb{R}]$ almost everywhere with linear combinations of those products, including all the solutions to the original equation.

Luckily, there are entire theoretical programs that were dedicated to precisely this, and you'll find that such theorems indicate the affirmative for the Schrodinger equation (that all solutions for the Schrodinger equation are separable when the Hamiltonian is time-independent).