The printing system in Chromium can support multiple simultaneous print jobs (one per CefBrowser instance). In order to implement this functionality in CEF on Linux it's necessary to differentiate the CefBrowser associated with each CefPrintHandler callback.
Multiple simultaneous print jobs should also be possible to implement on other platforms. However, the platform dialog implementation in Chromium cancels the additional print jobs. For example, in PrintingContextSystemDialogWin::AskUserForSettings on Windows the call to ShowPrintDialog returns an error when the print dialog is already displayed.