ProcessWire keeps a memory cache of pages loaded, so when you retrieve a page, its going to be the same instance of that page when you retrieve it elsewhere. But once you save a page (in the API) that memory cache gets cleared. If you wanted to keep an in-memory copy of it before it was saved, you'd want to clone it sometime before the page is saved:
$pageCopy = clone $page;
From that point forward, any changes to $page won't be seen in $pageCopy. This clone will cease to exist once it is out of memory scope.
For obvious reasons, file-based assets aren't cloned here. So if your clone depends on file/image assets, then you have to create a new page not just in memory, but on database/disk too. Here's how you do that:
$pageCopy = $pages->clone($page);
That does a recursive clone, cloning not just that page, but any children too. See the parameters to the $pages->clone() function for additional options to control this. You can also specify that you want it to clone to a different parent. Be careful with the cloning as it's something you probably don't want to do on every page save. It can be an expensive operation if there are lots of files or subpages associated with the cloned page.