Given a complete measure space $X$, consider the collection $F$ of full measure sets. These are the sets whose complement has measure 0.
Note that if $Y\in F$ then any superset of $Y$ (that is, any $Z$ with $Y\subseteq Z\subseteq X$) is in $F$ as well. (This is why I require the measure to be complete, which means that any subset of a measure zero set is measurable---and therefore of measure 0.)
Also, $X\in F,\emptyset\notin F$, and if $A$ and $B$ are in $F$ then so is their intersection. In fact, $F$ is closed under countable intersections.
This means that $F$ is a $\sigma$-complete filter. The members of $F$ are "big" (in the sense of the measure). Their complements are small (one even refers sometimes to measure zero sets as null or negligible). The sets of positive measure are thus those that are not small. They do not necessarily belong to $F$, but they are not null either.
It is in this sense that the notions are analogous: Club sets give us a notion of largeness (the analogue of full measure sets). Indeed, given $\kappa$ regular, the subsets of $\kappa$ that contain a club form a $\sigma$-complete filter (actually, a $\kappa$-complete filter). The stationary sets are those that are not disjoint from any club. Thus, they are not negligible in this interpretation.
Any reasonable filter will give you a similar analogous notion of largeness: large sets are those in the filter. Negligible, or small sets are their complements. Those that are not negligible play the role of stationary sets (or of sets of positive measure).
This is a common theme in analysis, where we also study the collection of comeager sets. Analysts study several other filters of sets as well. There is no universal notion of largeness, of course, what filter to consider is definitely context dependent. In infinitary combinatorics, the club filter has shown over and over again to be central.