Reaction rate ambiguity
I assume that the reaction rate must be the same (v), right? It should not change with stoichometry.


No, the reaction rates changes when you double all coefficients. Nothing real changes though (i.e. the rate of disappearing $\ce{H2O2}$ etc.).

You have a similar case with equilibrium constants: when you double or halve all the coefficients, the new equilibrium constant will be the square or the square root of the original one. The molar reaction enthalpy will also change when you change coefficients. It is a choice, and it still correctly describes the reaction.


  Any help?


There is no need for help. If you use either chemical equation format to calculate a measurable quantity (changes in concentration, equilibrium concentrations, enthalpy change for a specific amount that reacts, for the three topics respectively), you will get the same result. It is almost like the choice of describing a volume using liters or milliliters - it does not change the result even though it might look different to someone not used to the milli prefix..