The reaction between carbon dioxide and water to form hydrogen carbonate is an example of a bond-reorganization reaction. As one of the reactants is water it is also referred to as a hydration reaction. It is not an oxidation-reduction reaction.
The formation of carbon dioxide and water are redox reactions which involve the oxidation of carbon and hydrogen respectively with oxygen being reduced in both the case.
When the two combine, a bond between oxygen and hydrogen is broken and in each molecule of water with the oxygen bonding instead with carbon. As carbon can form 4 covalent bonds, it forms a double bond with one oxygen atom and a single bond with two other oxygen atoms.