Just like the page hierarchy, the Content Browser is geared towards the currently active website. You can switch to “All sites”, though, for selecting a piece of content from elsewhere, e.g. the image to use in an image widget.
As a default, when uploading a file, it initially gets associated with the current website, but you can change this assignment using the site selector in the properties of the asset. Select “Global” if you want the file to be available on all websites.
However, the site assignment of CMS objects that are part of the page hierarchy cannot be altered, meaning that a homepage and the pages directly or indirectly underneath it cannot be transferred individually to a different site as this would damage the site’s structure. You can, however, change the site assignment of pages as well as other resources not connected to the hierarchy, e.g., of landing pages.