Page numbers are simply fields. To go on multiple pages with one insertion they need to go in headers or footers. Every Section has three different headers and three different footers. The headers and footers in one Section may or may not be linked to those in the previous Section. One or more of these in your second section is apparently not linked to the first one. You need to go through every header / footer to make your changes.
Header/Footer Settings Recap

Again, headers and footers are different parts. It is confusing, more so than it needs to be because if you insert a different page number in a header it replaces the current header (including any other text you may have in the header). I see this replacing as a bug or design flaw. It led you to think the same thing would happen when you inserted in a different part. This is a reasonable assumption, just mistaken.