Construction of graphs where every pair of vertices have an unique common neighbor
If you get rid of the "no vertex of degree $n-1$" condition, the graphs with the property that every two vertices have exactly one common neighbor are exactly the friendship graphs (a set of triangles glued together at a common vertex); as the linked article describes, this is a theorem of Erdős, Rényi, and Sós. But obviously, all such graphs have a vertex of degree $n-1$; the only regular one is a triangle. So the answer to your question is that, no, a graph with the common neighbor property and without a degree-$n-1$ vertex does not exist.