“colors running” is an issue with dyes or paint not staying where it supposed to on something, so like washing a piece of clothing and the colors fading or staining some other part. Originally it references paint “running” off something in small streams, like if you applied to much. Running being a word commonly used to describe the flow of a stream or river, so if paint is coming off something in small streams, it would be described as “running”.

“The colors” also being a euphemism for a flag or standard on a battlefield, historically formations on a battlefield having a flag that would act as a clear point to focus around during the chaos of a battle, hence the saying “rally around the colors” meaning “ find the flag of the unit and defend it, if you don’t know what to do”.

So shirts with a flag that say “these colors don’t run”. it’s a double entendre, saying that “those that this flag stands for will not run away from a fight” and that “the dye on this clothing will not get distorted”.

In the context of a trans flag, it is a parody of vocally/excessively patriotic people who would wear an American flag shirt with that saying on it.

I think it’s in the sense of: they don’t run away and hide, but rather proudly stand and fight against prejudice.