If you're using one of the cover pages from Insert > Cover Page, then the "text box" isn't a text box or a Shape. It's a content control containing the built-in document property named Title. That document property is mapped to XML in the document file.

You can repeat the content anywhere else in the document, including in a header, by placing the cursor at the desired location and clicking Insert > Quick Parts > Document Property > Title. The same is true of any of the other document properties in that list.

If you edit the content of one Title content control. the change will automatically appear in all other Title content controls anywhere else in the document.