How to explode set of lists into subsets under criterion?

"Overlapping" is not an equivalence relationship because it is not transitive. So approaches using Gather and related functions are not possible.

The simplest, though not the most efficient way is to test every pair of elements from the set. After doing that, we can treat this as a ConnectedComponents problem.

Code:

set = {{a, {1, 2, 3}}, {b, {1}}, {c, {2, 3}}, {d, {4, 5}}, {e, {5}}};

am = Outer[Boole@IntersectingQ[#1[[2]], #2[[2]]] &, set, set, 1]
(* {{1, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 1, 0, 0}, {0, 0, 0, 1, 1}, {0, 0, 0, 1, 1}} *)

components = ConnectedComponents@AdjacencyGraph[am]
(* {{1, 2, 3}, {4, 5}} *)

Part[set, #] & /@ components
(* {{{a, {1, 2, 3}}, {b, {1}}, {c, {2, 3}}}, {{d, {4, 5}}, {e, {5}}}} *)

---

Addendum

This is what could happen if we try to use Gather anyway:

sets = Partition[Range[5], 2, 1]
(* {{1, 2}, {2, 3}, {3, 4}, {4, 5}} *)

Gather[sets, IntersectingQ]
(* { {{1, 2}, {2, 3}}, 
     {{3, 4}, {4, 5}} } *)

ConnectedComponents @ SimpleGraph @ RelationGraph[IntersectingQ @@ {##}[[All, 2]]&, set]

{{{a, {1, 2, 3}}, {b, {1}}, {c, {2, 3}}},
{{d, {4, 5}}, {e, {5}}}}