Coloring where all colors are present in closed neighborhood of every vertex
I would call this a polychromatic coloring of the closed neighborhood hypergraph.
I don't think this has been studied before for general graphs.
Here is a paper studying the question when edges are colored:
Béla Bollobás, David Pritchard, Thomas Rothvoß, Alex Scott: Cover-Decomposition and Polychromatic Numbers
And here is one that studies the conflict-free coloring problem for your hypergraph:
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew: Conflict-free coloring on closed neighborhoods of bounded degree graphs
If you replace closed neighborhood with geometric range spaces, then you can find a lot more literature, most notably my own amazing papers.