he said system registration is only necessary on entrance and not on Exit
If he actually said "system registration", he was wrong: there is no Schengen-wide "system registration" at all. When they scan your passport they just get a digital copy of your passport data along with possible alerts (such as arrest warrants) about the passenger, and they are supposed to do that on entry to the Schengen Area, and optionally on exit, though even on entry it's often not strictly followed, which really means nothing to you as a traveller.
Some countries have systems for their own use (which is not visible to the others), but it's still only the passport stamps that are the important thing. That's why you, as a non-EU/EFTA visitor, always have to make sure to get stamped when entering and exiting Schengen.
Sometimes, border agents (most commonly in France and Italy) will straight up lie and say it's "not needed" for "low-risk" passports (American, Australian etc.) because lifting their hand for a second seems to drain them of all life energy. They are breaking the law, however (specifically the Schengen Borders Code), by doing that, and you should never let them as you could get to pay for it.