I am not sure how does removing cut edge {a,b} disconnects the graph. My interpretation of disconnect means when the graph has multiple components.
Original graph before removal
graph after removal-how is this a graph that is disconnected? It should not be disconnected, correct?
Removing an edge means deleting only the edge, and not the vertices incident to it. In your example, the vertex $a$ remains as an isolated vertex and isolated vertices are certainly connected components.