Consider a fixed model $M$ of ZFC or ZF.
A "set" $s$ in $M$ is a single element of $M$'s universe. One can identify a collection of sets (i.e. collection of elements of $M$'s universe, as seen from outside of $M$, in the meta-theory) with $s$ by collecting all sets that are contained in $s$ according to $\in_M$. A "proper class" is a collection of elements of $M$'s universe (outside of $M$, in the meta-theory) which is identified by some formula, but not identified by any set of $M$.
Following this pattern, one can introduce the concept of "proper collection" as some collection of elements of $M$'s universe which is not identified by any formula.
I would argue that proper collections have to exist in a model that is large enough in the meta-theory.
Are there any interesting theorems that refer to proper collections?
Look at forcing (over a separative partial order). A generic filter cannot be defined in the ground model by any formula (even with parameters), so this is an example of what you’re calling a proper collection.
Another example is $0\#$; if it exists, it’s a proper collection of $L.$