Zaitsev was a real sniper, that much is not disputed.
There was, however, no such German sniper found in any German records after the war. The sniper school mentioned in the story also didn't exist. That's why it is said to be false, or just Russian propaganda.
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34th SS-Freiwilligen-Grenadier-Division Landstorm Nederland
"Meine Ehre heißt Treue"