O=Cl=O and O-Cl-O in a ring are not resonance structures?

Resonance structures are about moving electrons to form anions, cations, and new bonds. Structural isomers can be resonance structures.

For example take phenyl ketone. Of course you can form resonance structures by moving the pi electrons from benzene. However you can also move the electrons from the carbonyl double bond to the oxygen forming phenyl oxide. From there you can again move the electrons from the benzene ring. All of these are resonance structures and the ones with the alkoxide anion are structural isomers as well as resonance structures.

In the same way you can move the pi electrons from the $\ce{O-Cl}$ double bond to the oxygen to form $\ce{O-Cl-O}$(I don't know how to make the negative charge on the left oxygen not look like a 4 e bond(which unless you are talking about a double bond is nonexistent)) which can then form a peroxide bond($\ce{O-O}$ single bond) to make a cyclic peroxide. Resonance structures are all about moving electrons and in the process forming anions within molecules(like the alkoxide anion), cations within molecules, and new bonds.

So how are chlorine dioxide($\ce{O=Cl=O}$) and this cyclic peroxide that forms not resonance structures and just structural isomers?

Answer

Resonance structures are about moving electrons to form anions, cations, and new bonds. Structural isomers can be resonance structures.

No, resonance structures do not form anions or cations. Resonance structures should all have the same overall formal charge.

Also moving electrons around when drawing resonance forms does not create new bonds. These bonds always have existed; it's just that one picture is inadequate to represent them all. Hence the resonance forms.

Structural isomers also cannot be resonance structures. Structural isomers differ fundamentally in terms of the bonding between or among their constituent atoms. Resonance structures are just things we draw on paper to better represent the true nature of bonding. For one chemical entity, there may be multiple resonance forms, but that doesn't mean that each is a discrete form that is able to be isolated.

In the same way you can move the pi electrons from the O−Cl double bond to the oxygen to form O−Cl−O(I don't know how to make the negative charge on the left oxygen not look like a 4 e bond(which unless you are talking about a double bond is nonexistent)) which can then form a peroxide bond(O−O single bond) to make a cyclic peroxide. Resonance structures are all about moving electrons and in the process forming anions within molecules(like the alkoxide anion), cations within molecules, and new bonds.

So how are chlorine dioxide(O=Cl=O) and this cyclic peroxide that forms not resonance structures and just structural isomers?

I'm not sure what you mean by this.