Unlike Calc, in Writer a sheet is not something self-contained. Writer document defines page styles, in them it defines page style order, and that’s all. The main entity in Writer is contents, which is dynamically laid out on created-on-demand pages of corresponding types. Specific parts of contents (namely, some paragraphs and tables) define that they must start on new pages [possibly with specific styles]. But otherwise, page is highly dynamic concept. So, in electronic Writer document, it’s unwise to think about page as if it’s printed sheet of paper.
Thus, writer has no means to copy page. Instead, it allows you to copy contents. If it happens to start with a paragraph that defined page break, then it will arrive on new page in destination document (provided that destination application understands required attributes). And if destination document’s page style differs from source (and the source style hasn’t been copied with contents), then the contents will be laid out differently
You also should understand that contents may contain dynamic contents itself, like fields. So, even the text in destination document may differ from source, e.g. if field contents depends on previous data (like, say, pages number).
Just to make the answer seem relevant, I mention that you need to select the range that you want to copy, and use “Copy” button, or menu Edit->Copy, or Ctrl+C.