Whether it's made by microbes or industrial processes, the dye is still chemicals. Every physical object is chemicals. The microbes are made of chemicals and make chemicals.
Probably more importantly though, there's no particular reason the chemicals made by bacteria should be safer or less impactful to the environment.
There's a few reasons to think that would tend to be the case, but any given compound isn't magically safe for human exposure because a micro-organism synthesized it.
That's really not true though. The downstream processes after biosynthesis will have steps like "lyse all the cells open" and ,"definitely and thoroughly kill them and the off target proteins".
Like you're going to do a lot of very specific, biotoxic things.
It has water as the solvent so plenty of techniques here that dont involve toxic things such as rapid heat pasteurization, as is done in milk processing. Also see centerfugation, crystalization, filtration, etc. You can lyse all cells with shear force easily by mixing excessively fast.
There's a few reasons to think that would tend to be the case, but any given compound isn't magically safe for human exposure because a micro-organism synthesized it.