> Does Herodotus' characterisation sound rooted in Zoroastrianism?
My assumptions: The Zoroastrian Persians of that time were "Zoroastrian" like today's Christian Americans are "Christian".
> educate their boys from five to twenty years old,
Based on how me, my family, and friends were all raised, yes education is paramount. It falls into "make life a positive sum game" or "grow". That seems fundamentally Zoroastrian.
The exclusion of women strikes me as anti-Zoroastrian. Maybe this is a more modern ethos even for Zoroastrians. Every Zoroastrian, regardless of sex/gender, that I know is educated at least till 20, and past that if they desire it/finances accamodate).
> and teach them only three things: riding and archery and honesty
Making it exclusive to riding, archery and honesty seems non-Zoroastrian. Likely just pragmatic for their army or something. Even that seems off to me - they had farmers and mathematicians and engineers, etc. If I were guessing this is selection bias from likely his Greek roots, the ongoing wars, and living in a smaller city (I assume closer to the borders of the empire).
The honesty does resonate, but its likely an incomplete translation. I'd assume they actually meant "noble"[0], which would be rooted Zoroastrianism. Where specifically they taught honesty as important, but caveated with pragmatism to withheld to ensure "high moral outcomes". "High moral outcomes" is very rooted in Zoroastrianism.
[0] having or showing fine personal qualities or high moral principles and ideals.