Why is H2O2 named hydrogen peroxide?
Oxide is a term reserved for the oxide anion $\ce{O^2-}$. This anion is typically characterised as rather stable since the very electronegative element oxygen is in its maximum negative oxidation state. Dioxide means, that there are two oxides present in whichever structure we are looking at.[1] A typical example for a dioxide would be manganese dioxide, also called manganese(IV) oxide $\ce{MnO2}$. As shown in the image below, it crystallises in the rutile structure, like titanium dioxide $\ce{TiO2}$.


Figure 1: Crystal structure of rutile, $\ce{TiO2}$ which is equivalent to &#x3b2;-$\ce{MnO2}$. Red spheres represent oxides, grey ones the metal in question. Image taken from Wikipedia, where a full list of authors is available.

Hydrogen peroxide, on the other hand, as is evident from the structure in figure 2 below, contains not two $\ce{O^2-}$ units but a single $\ce{O2^2-}$ unit, composed of two oxygens bound to each other. This means that oxygen’s oxidation state is $\mathrm{-I}$ rather than $\mathrm{-II}$ and that hydrogen peroxide is readily reduced. (It can also easily be oxidised back into $\ce{O2}$. The functional group $\ce{O-O}$ is labelled peroxide and it would not make sense calling it dioxide since it does not feature oxides.


Figure 2: Structure of hydrogen peroxide. Red spheres are oxygen, white ones hydrogen. Image taken from Wikipedia, where a full list of authors is available.



Aside from that, IUPAC also allows for nomenclature to entirely ignore the molecular structure and name entirely by molecular composition. In that compositional nomenclature, dihydrogen dioxide is an acceptable name like dihydrogen monooxide would be for water. Note, however, that this only tells us the number and type of atoms involved while not giving us any clue to how they are connected. It is thus typically better to refer to a peroxide as a peroxide to emphasise the actual functional group present.