I think that you have made a very astute observation!
I also think that you probably use some kind of bleach, or oxidising cleanser, in your toilet!
How do I know?
Well, blood is red because it contains haemoglobin, which is the molecule that ferries oxygen from the lungs to our tissues. As the name suggests, each molecule of haemoglobin contains an atom of iron at its centre in a structure called a porphyrin ring.
Like a number of metals, iron has more than one oxidation state. Commonly it donates 3 electrons to make Fe3+ ions (like rust - Fe2O3), but it can also donate just 2, forming Fe2+ ions (like iron II sulphate FeSO4).
The iron in haemoglobin is in the 2+ oxidation state (Fe2+), and this is critical because if the haemoglobin is oxidised (to "methaemoglobin" containing Fe3+) then it can no longer work as an oxygen carrier. Some drugs and toxins can do this.
In the case of your blood-soaked tissue, I suspect that when you threw it into the toilet the toilet water contained something like bleach, which is a strong oxidising agent (it contains hypochlorite). Perhaps you've got one of those little tablets that sits in the cistern or dangles beneath the rim of the toilet and dissolves slowly with every flush?
Either way, these chemicals would have oxidised the Fe2+ in your haemoglobin to Fe3+, altering the colour (methaemoglobin is brown, like rust), hence your observation!