A technique improves efficiency of a copy-on-write (COW) operation used to create a snapshot and/or clone by a volume layer of a storage input/output (I/O) stack executing on one or more nodes of a cluster. The snapshot/clone may be represented as an independent volume, and embodied as a respective read-only copy (snapshot) or read-write copy (clone) of a parent volume. Volume metadata managed by the volume layer is organized as one or more multi-level dense tree metadata structures, wherein each level of the dense tree includes volume metadata entries for storing the metadata. The volume metadata entries may be organized as metadata pages having associated metadata page keys. Each metadata page is rendered distinct or “unique” from other metadata pages in an extent store layer of the storage I/O stack through the use of a multi-component uniqifier contained in a header of each metadata page. To improve the efficiency of the COW operation, the technique allows the use of reference count operations on the metadata page keys of the “unique” metadata pages so as to allow sharing of the metadata pages individually between the parent volume and the snapshot/clone.