The Fubini-Study metric on $\mathbb P^n$ arises as the curvature of a line bundle. More precisely, once you fix a hermitian inner product in the complex vector space you are projectivizing, you get a natural metric on the tautological line bundle $\mathcal O(-1)$ on the projective space. If you take the dual metric on $\mathcal O(1)$ and you compute its Chern curvature you find the Fubini-Study metric.
Once you look at things in this way, on the Grassmannian you have a completely analogous construction (which, after all, gives you the induced metric of the Plücker embedding in a somewhat more direct way).
Call $S\to G(k,n)$ the tautological holomorphic vector bundle of rank $k$ (the fiber over each point is the $k$-dimensional subspace of the vector space $V$ you are considering). Then, once you are given a hermitian inner product on $V$, you can endow naturally $S$ with a smooth hermitian metric. In particular, you can induce a smooth hermitian metric on the holomorphic line bundle $\det S^*\to G(k,n)$. Now, compute its Chern curvature to discover that it is a (strictly) positive $(1,1)$-form, so that it gives you a hermitian metric on $G(k,n)$.
Observe that the map given by the complete linear system associated to $\det S^*$ is nothing more than the Plücker embedding, so that the metric we constructed here is what you were looking for, but given in a more intrinsic way.
Notice finally that this is really a generalization of the Fubini-Study metric, since when you look to $G(1,n)$, then $S=\mathcal O(-1)$ and the construction is as above.
There are of course more general constructions for complete or incomplete flag varieties, but the guiding philosophy is again the same: to put a natural hermitian metric on some (positive) line bundle and then to take the metric on your space to be its Chern curvature. For more detail details and computation computations you can look for instance at this paper by J.-P. Demailly.