Why is the solution to the Blasius boundary layer problem self-similar?

The wiki page on Blasius boundary layers is a useful and thorough resource in this case.

Blasius boundary layers arise in steady, laminar 2D flow over a semi-infinite plate oriented parallel to the flow. In this scenario, the Navier-Stokes equations are particularly simple and amount to a leading-order balance between inertia and viscous forces. The similarity solutions are derived from scaling arguments of these equations. It's not worth me repeating the derivation of the ode as the wiki page is very thorough.

The self similarity condition is applied when deriving the Blasius equation, implying that the streamwise velocity $u(x, y)$ across a boundary layer is a function of the derivative of $f(eta)$ scaled by the flow $U$ as follows $$u(x, y) = Uf'(eta)$$ The self-similar streamwise profile derives from the following stream function. $$psi(x,y) =(Unu x)^{frac{1}{2}}f(eta)$$ To see this clearly, we differentiate the streamfunction with respect to $x$, finding the $x$ component of the velocity. $$u(x,y) = frac{partial psi}{partial y} = U(Unu x)^{frac{1}{2}},,f'(eta)frac{deta}{dy} = U(U nu x)^{frac{1}{2}},,f'(eta)bigg(frac{U}{nu x}bigg)^frac{1}{2}$$ Which simplifies to the desired equation.

If there are no solutions of that form you wont find any.