Since 2017 or maybe 2018 I haven't gotten around to re-reading the revised Schengen border code again; but, IIRC, at least in 2017 (or maybe even 2018, if not even more recently) EU countries' national entry and exit registers weren't required by the Schengen border code (or anything else at the supra-national level) to register EU nationals' Schengen entries and/or exits.

The OP's child coming into the Schengen zone on an Italian passport may have had the entry registered nowhere, even if entering the Schengen zone in Italy or elsewhere, and that would have been fine under the Schengen border code. Whether it's fine with national border codes or not.