Shared Elements Algorithm
Construct an undirected graph with $k$ vertices, where each vertex represents a set $A_i$, and you draw an edge between two vertices if their sets share an element.  Now you have an instance of subgraph isomorphism: you have a graph $H$ on five vertices (of the shape shown in your question), and you want to enumerate all subgraphs of $G$ that are isomorphic to $G$.  You do have an additional distinctness condition, but if you can enumerate all isomorphic subgraphs, you can check whether they satisfy that additional condition in post-processing (and in most cases I expect they typically will).
This subgraph isomorphism problem looks challenging.  A naive approach has $O(k^5)$ running time: enumerate all 5-tuples of vertices.  You can improve this slightly, to $O(k^2m^{1.5})$ time, where $m$ counts the number of edges in the graph (note that $m\le k^2$, and might be much smaller, so this might be better).  First, enumerate all 3-cliques (triangles).  This can be done in $O(m^{1.5})$ time (see http://complexnetworks.fr/wp-content/uploads/2011/01/triangles.pdf and https://users.soe.ucsc.edu/~sesh/pubs/conf-triangle-enum.pdf).  These will be your candidates for $A_1,A_2,A_3$.  Next, enumerate all possible candidates for $A_4,A_5$ and check to see if they meet the requirements.  This might not be much faster, but maybe it would help a little bit.