[Recently, I answered your question and relevant information can be found from that answer.][1]

At first, I thought that the brown colour is due to ferrous chloride that formed during the course of reaction but since the reaction is occuring in a aqueous solution, it should be green in colour. Ferrous chloride is brown in anhydrous form. So I thought to be either due to formation of a complex, [iron(II, III) oxide($\ce{Fe2O3}$),][2] or [iron(III) oxide-hydroxide($\ce{FeOOH.nH2O}$)][3] or a mixed salt of oxide-chloride i.e. [iron oxychloride($\ce{FeOCl}$).][4]


The following information is from the paper 2 of that answer:

> According to the mechanism proposed by KREMER and STEIN, **an
> intermediate oxygen complex of iron with oxidation number +V is
> primarily formed by the reaction of $\ce{Fe^3+}$ with $\ce{H2O2}$.
> This complex reacts with another $\ce{H2O2}$ molecule to water and
> oxygen thereby reforming $\ce{Fe^3+}$.**
> 
> $$\ce{Fe^3+ + H2O2 <=> [Fe^{III}OOH]^2+ + 2H+ <=> [Fe^{V}O]^3+ + H2O ->[H2O2] Fe^3+ + 2H2O + O2}$$
> 
> According to the mechanism proposed by HABER and WEISS the $\ce{Fe^3+}$ ions
> initiate a radical reaction, after which the chain reaction consumes
> the hydrogen peroxide. This mechanism can explain the high reaction
> rate very well.
> 
> Chain initiation:  $\ce{Fe^3+ + H2O2 <=> [Fe^{III}OOH]^2+ + 2H+ <=> Fe^2+ + HOO. + H+}$
> 
> Chain propagation: $\ce{Fe^2+ + H2O2 -> Fe^3+ + 2OH.}$ 
$\ce{Fe^3+ + H2O2 + OH. -> Fe^3+ + HOO. + H2O -> Fe^2+ + H+ + O2 + H2O }$

  
That intermediate oxygen complex of iron with oxidation number +V maybe responsible for the brown colour of solution.


----------
As for the potassium iodide, the color you are seeing is basically food coloring. Various food coloring  is used to make the foam produced colorful. If food-color is not used, still color is seen. That color is basically color of pure soap foam. [This elephant toothpaste experiment not uses food-color but still it has color that is color of soap-foam.][5]


  [1]: https://chemistry.stackexchange.com/questions/66949/demonstrating-decomposition-of-hydrogen-peroxide-using-ironiii-nitrate-catalys/67059#67059
  [2]: https://en.wikipedia.org/wiki/Iron(III)_oxide
  [3]: https://en.wikipedia.org/wiki/Iron(III)_oxide-hydroxide
  [4]: https://en.wikipedia.org/wiki/Iron_oxychloride
  [5]: https://www.youtube.com/watch?v=UTfMrx7275w