Studying some problem I've arrived to the following notion. 


Let a $2r$-regular graph $G$ be called *neighbour-matching* if $N(v) = rK\_2.$ In other words, the neighbourhood of any vertex induces a matching in $G.$


Doing some computation, it appears that such graphs are quite rare. There is one such graph on 9 vertices (generalized quadrangle) and two such graphs on 12 vertices. An interesting example of such a graph is [Brouwer–Haemers graph](https://en.wikipedia.org/wiki/Brouwer%E2%80%93Haemers_graph) of order 81.


I am interested in the following (hopefully not obvious) questions


1. Is the notion of such graphs already known under some other name?
2. Is there a nice characterization of such graph? Do they have any interesting properties?



Regular graphs whose neighbourhoods induce matchings
This answer just suggests a couple of directions that you can continue searching, rather than being definitive.


Brouwer, Cohen and Neumaier use the term *edge regular* for the property that every pair of adjacent vertices have the same number of common neighbours (in other words, like strongly regular but dropping the condition on non-adjacent vertices). So your property is then edge regular with $\lambda = 1$. Of course any graph that is actually *strongly regular* with $\lambda = 1$ will do the trick, hence the Brouwer-Haemers graph and the Paley graph on 9 vertices that you already found. There are other SRGs with $\lambda=1$. There are various papers on edge-regular graphs of different types, but I don't know offhand if $\lambda=1$ has been tackled.


On another tack, a graph is called *locally $X$* if every neighbourhood is isomorphic to $X$, and so you are looking for graphs that are locally $rK\_2$. Again there is lots of work on graphs that are locally this, locally that and locally the other, but I don't know offhand for $X =rK\_2$.


A third direction in which you may search is to observe that your condition is equivalent to every edge being in a unique triangle (+ regularity). This sounds like a condition that somebody should have studied, but at the risk of sounding like a broken record, I don't know offhand of any such results.


However I think that there is definitely scope there for some interesting work.


