Denticity of bridging ligands
It is important to clearly separate the terms denticity, hapticity and bridging.

An atom or a ligand is bridging is one single atom bonds to two different acceptors.

A ligand (but never a single atom) can be multidentate or multihaptic if two atoms of that ligand can bond to the same acceptor.


A ligand such as the amide anion cannot be multidentate or multihaptic as it only has one coordinating atom: nitrogen. This nitrogen can coordinate to two different metals (in a μ-amido complex) but it still remains a monodentate ligand.
A ligand such as ethylenediamine or en for short ($\ce{H2N-CH2-CH2-NH2}$) can coordinate to the same central atom via each of the two nitrogen atoms. This is a bidentate ligand as it has two ‘teeth’ in the central atom.
A ligand such as ethene ($\ce{C2H4}$) can coordinate to a single central atom in a bihaptic (η2) fashion. Drawing shortest lines from nucleus to nucleus, both carbon atoms would be the same distance from the acceptor atom (in reality, it is better thought of the π system donating). Hapticity is reserved for such cases in which a continuous set of atoms coordinate to a central metal.
A ligand can be both multihaptic and multidentate. An example would be dibenzylideneacetone or dba ($\ce{Ph-CH=CH-C(=O)-CH=CH-Ph}$), in which both of the $\ce{C=C}$ double bonds act as a bihaptic ligand. (Due to geometry constraints, dba usually coordinates to two different acceptors rather than to a single one in a chelate fashion.)
Finally, it is possible for a ligand such as en to be both bidentate and briding, if either of the nitrogen atoms coordinates to two different metals. This requires deprotonation of at least one nitrogen.
The bottom line is: briding ligands can be monodentate or multidentate; they can be monohaptic or multihaptic. Only bridging ligands based on a single donor atom must be monodentate.