Sakurai mentions (in various editions) that the propagator is a Green's function for the Schrodinger equation because it solves $$\begin{align}&\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) \cr= &-i\hbar\delta^3(x-x_0)\delta(t-t_0).\end{align}\tag{2.5.12/2.6.12}$$
I don't see that. First of all, I don't understand where the $-i\hbar$ Dirac delta source term comes from.
And if I recall correctly, a Green's function is used to solve inhomogeneous linear equations, yet Schrodinger's equation is homogeneous $$\left(H-i\hbar\frac{\partial}{\partial t}\right)\psi(x,t) = 0,$$ i.e. there is no forcing term. I do understand that the propagator can be used to solve the wave function from initial conditions (and boundary values). Doesn't that make it a kernel? And what does Sakurai's identity mean?