I suggest that you move over from the manifold point of view, to use the refined approach of algebraic geometry. This is because the Plucker embedding is actually "algebraic".

In that setting, Grothendieck's Quot scheme would classify quotients(or, looking at the kernel, subbundles) of a free vector bundle. Actually, we use locally free sheaves instead of vector bundles. Indeed, this Quot scheme is stratified for each polynomial, classifying the quotients with a certain "Hilbert polynomial".

And then the Grassmannian is just a particular component of the Hilbert scheme, corresponding to just one particular Hilbert polynomial. The construction of this particular case is precisely via the Plucker embedding, realizing the Grassmannian as a projective variety. On the other hand, in the construction of Quot scheme I have read, it is constructed as some subscheme of the Grassmannian. In this sense, the Plucker embedding is very intertwined with the study of vector bundles.

(Here one might note that this answer is somewhat similar to Francesco Polizzi's).

I note that it is possible to move over from the manifold point of view, to use the more refined approach of algebraic geometry. This is because the Plucker embedding is actually "algebraic".

In that setting, Grothendieck's Quot scheme would classify quotients(or, looking at the kernel, subbundles) of a free vector bundle. Actually, we use locally free sheaves instead of vector bundles. Indeed, this Quot scheme is stratified for each polynomial, classifying the quotients with a certain "Hilbert polynomial".

And then the Grassmannian is just a particular component of the Hilbert scheme, corresponding to just one particular Hilbert polynomial. The construction of this particular case is precisely via the Plucker embedding, realizing the Grassmannian as a projective variety. On the other hand, in the construction of Quot scheme I have read, it is constructed as some subscheme of the Grassmannian. In this sense, the Plucker embedding is very intertwined with the study of vector bundles.

(Here one might note that this answer is somewhat similar to Francesco Polizzi's).

Added note(prompted by BCnrd's comments below): If you are not into algebraic geometry, please forget completely about this post. If you ever get into it, then you might come back and read it again.