How do we derive Lift Coefficient of an airplane's wing?
The lift coefficient accounts for complex aerodynamic effects that are impossible to calculate by hand in any practical way. I'm not sure what you mean by "In my opinion, even it correct mathematically, but is not correct in the engineering as the it is not product of mathematic." My best guess is that you're pointing out that even though you can rearrange the lift equation to calculate $C_L$, it's not derived from first principles, which is correct. It's a convenient way to represent more complex interactions that are difficult to calculate.

Essentially, we know lift varies linearly with dynamic pressure ($\frac{1}{2} \rho V^2$) and it varies linearly with surface area. So the lift equation may be written, in its most reduced form,
$$L = C_L q S.$$
Notice the similarity to the drag equation:
$$D = C_D q S$$
And the pitching moment equation:
$$M = C_M q S c$$

Each of these quantities varies linearly with dynamic pressure and surface area, so it's useful to express them as products of these quantities. Changing either of these variables will probably affect the aerodynamic coefficient as well, but for modeling purposes that don't require extreme precision, in most cases, the aerodynamic coefficient won't change nearly as much as dynamic pressure can.

As a result, knowing $C_L$ and $C_D$ for a variety of flight conditions, you can model your body of interest with fairly good accuracy without having to go through the complexity of computational fluid dynamics, which is the numerical way to predict aerodynamic forces.

To answer your question about a formula to calculate the coefficient of lift, there are methods that will allow you to get a rough estimate (e.g. lifting line theory), but to get results comparable to wind tunnel testing, you have to use computational fluid dynamics, which will require a computer to achieve meaningful results.