In Algebraic Geometry, there is this nice interpretation for the localization of rings and for quotient rings. Let $A$ be a ring. Then, localizations of $A$ at different elements correspond to the distinguished open subsets of Spec$(A)$, and quotients of $A$ over different ideals correspond to closed subschems of Spec$(A)$. Thus in broad terms, it is helpful to think of localization as opens, and of quotients as closed.
This interpretation helped me understand why many properties of rings work well with localization but do not with quotients (e.g. being a UFD). This makes sense to me because we usually understand the elements in $A$ as functions on Spec$(A)$, and functions behave better on open sets than on closed sets.
However, I have trouble understanding Noetherian rings with this intuition, as this property works well both with localizations and quotients. Is there any geometric intuition behind why Noetherian rings work well with both open and closed sets?
This may not be entirely what you're after, but I think a good (geometric) way to think about Noetherian-ness is that it's a very nice finiteness condition on the space $Spec(R)$. Many finiteness conditions on topological spaces pass to open and closed subsets, e.g, being a finite diameter metric space.
Examining the definition now, translating the algebraic definition to topology gives the notion of a Noetherian space, one in which any descending chain of closed subsets stabilises. This clearly passes to both opens and closed subsets, and this finiteness condition is robust, commonplace (for $Spec(R)$), and incredibly powerful. For instance, one can prove that any Noetherian space is uniquely a union of its irreducible components, which in turn implies structural theorems about ideals in Noetherian rings.
While you're still coming to grips with the Noetherian property, I would think of it as a very convienent (and not too restrictive) finiteness condition, with many fantastic technical consequences, but of a slightly different flavour to finiteness conditions you might've seen before.
In general, "Noetherian-ness" is a widespread kind of finiteness condition which just happens to be first encountered in commutative algebra.