The vector $\xi$ is tangent to the complex line spanned by $\xi$. The significance of "infinitesimal" is that this definition defines a length of a tangent vector (in the sense of differential geometry), which is thought of traditionally as a point infinitesimally near to $P$. There is another definition, which you should find in the same book, of a metric, which will actually give distances between points of $\Omega$ (at least, say, for andany bounded domain $\Omega$ in complex Euclidean space). The length is not measured from the origin to $P$, but from the origin of $T_P \Omega$ to $\xi$, i.e. it is the length of a tangent vector, similar to how one measures lengths of tangent vectors in a Riemannian metric. More precisely, the infinitesimal Kobayashi length is a seminorm on each tangent space. It might help to free your intuition if you forget that $\Omega$ is a domain in a complex vector space, and define the infinitesimal Kobayashi length on any complex manifold $\Omega$. There is no reason why $\Omega$ should contain any origin, or have a meaningful notion of distance from any origin.
