Dipping into the history of ink has been somewhat disappointing in that there is not much history about the ‘color’ of writing ink. How did black, blue and red ink come to dominate the ink for handwriting today?
The Egyptian and Chinese cultures invented ink simultaneously using charcoal or soot and animal fat hence black ink was used for the early written word. There are plenty of perfunctory stories about carbon ink [from soot], iron gaul / oak gall ink [which starts as a bluish-black and fades to dull brownish black] and the evolution of printing ink [mostly an oily black varnish-like ink that would not smudge]. Ink used for writing has two main components – Colorants & Carrier/Binders. Pigment and dye have both been used as colorants
I did find reference of Scribes represented with multiple inkpots in the Middle Ages, suggesting several colors at their disposal when copying texts.

The three colors found most frequently where black from oak galls, red from red lead and crushed lapis lazuli creating a valuable and rich shade of blue. These are also the three colors we see today in pen choices.
So that’s the story, or all I can find. We have primarily Black, Blue and Red writing ink … ‘because we’ve always done it that way’ … since the Middle Ages!
POSTSCRIPT: There is an ink color & smell you are bound to remember if you attended grade school in 1950s – 1970s… that of the mimeograph or ditto machine

I remember the mimeograph it was the ‘copy machine’ of my grade school years. The ink ended up looking like a faded blue or purple AND the printed page had that intoxicating fragrance! That unforgettable fragrance imprint came from the duplicator fluid which had methanol and isopropanol in it! I can still smell the ink on the warm pages…..