Since we can set the world coordinates ourselves, we can let the world $Z$ coordinate be $0$, discarding the third column of the matrix $P$, resulting in a matrix we call $H$.
$H$ has $9$ entries, but accounting for $\lambda$, there are only $8$ parameters. Thus, we need four points to determine $H$.
Homography works for image stitching because it helps us map different camera angles onto the same coordinate plane.