Upvote:3
I would second everybody else's recommendation not to take the rules lightly and consider possible alternatives. What you are proposing is unambiguously illegal. Checks are not always very thorough and the consequences of an overstay vary a lot but they are potentially serious enough (significant fine, multi-year ban) to be wary of taking this risk.
But to answer your question directly, contrary to what many people believe, enforcement in the Schengen area does not rely primarily on a shared database of all entries and exits. To the best of my knowledge, such a database does not presently exist but there are plans to roll one out (the Schengen Entry and Exit System or EES), with the ambition to reach full interoperability in 2020. That said, there are still many ways your plan could backfire. I think PNR data is already being collected and I read that data will be kept for five years in the new system. Your niece's movements could be recorded in national databases too.
And the trip to Italy is not the only one that matters. Imagine she re-enters Austria two or three years later with the passport that now bears traces of the long trip to Italy. There is nothing preventing an Austrian border guard from starting a procedure based on the earlier overstay even if Italian border guards didn't notice originally. Not particularly likely but entirely possible.
Upvote:5
She should apply for a student Schengen visa and if that Art History program is an education program she should have no problem getting one (especially as a US citizen).
I would NOT recommend to try and bend the visa rules or trick the customs - you can easily end up getting banned form Schengen zone entry for a prolonged period of time if they suspect malice, or simply rejected entry if over 90/180.
Student Schengen visa (or maybe even national student visa if they offer one) to Italy is the way to go.
Upvote:10
Countries are not so easy to fool by getting a new passport. (The difference between "new" and "renewal" here is entirely internal to the forms required by the USA Passport Office; a renewal passport also has a new number.)
Perhaps she can arrange a student visa for the 45-day program. Normally it would need to be a longer program, but… Time spent in Italy on such a visa does not count towards 90/180. In fact, if she wants an entire gap year, perhaps she could continue at the same institution as the 45-day Art History program, which (off-topic) sounds like hella fun.