I think you might be a little confused around some admittedly inconsistent terminology. There never was a Kingdom of Savoy, rather there was a Kingdom of Sardinia ruled by the House of Savoy.

It's true that the County of Savoy, from which the eponymous house originates, was mostly in the French sphere of influence up until the 17th century, but pervasive "metropolitan" French influence outside of the upper class was negligible until the introduction of universal education in the 19th century (along with the arrival of mass media). Unlike what modern "hard" borders might suggest, languages and cultures naturally develop on a continuum, and there's nothing particularly special about the way historic language and culture of the Savoyard region is laid out: the local Provençal dialect is a Gallo-Romance language intelligible as far north as the Morvan Massif (more or less) and as far south as the Aosta valleyhead at Ivrea. It's true that the neighboring Italian dialects are fairly different than the dialects of the northwestern Alpine valleys, but they're still of the Franco-Provencal family and talking slowly and pointing a lot probably gets a Savoyard speaker understood as far east as the general whereabouts of Novara (although Novarese itself marks the transition to the Western Lombard subgroup) and as far south as the Ligurian foothills along the Cuneo-Alba-Alessandria line. I actually wrote about how language continuums might look like before modern borders in this answer from a month ago which you might be interested in. The major take away, I suppose, is that it is common for people to have all sorts social and familial ties on both sides of a border.

On the topic of "French Influence" in Italy, there is indeed something to be said about Napoleon's influence in Italy, but I get the feeling that's not really what you're asking for. In any case, I wrote a fairly long answer about the role of the "Savoyard State" in the unification of Italy, and it's relationship with France and especially Napoleon, in this answer from about a year ago which you might be interested in. I also wrote this much shorter answer about naming conventions for the Savoyard State which might also clarify some things.