Generally putting things in score makes it appear in parts, and most changes to parts are by design made local, so You can have part-specific adjustments. I believe, and I could be very wrong, but frames are also local in score, as the Part inherits all music and variables, but not frames, as they are not connected to each other in any way, which is also by design, to allow freedom in expense of simplicity of some workflows.
To reuse frame layouts You should apply master pages, accessible from Engraving tab, right sidebar. You can make Your own.
Unless the text frame is in a master page, it’s a layout frame and therefore only appears on the page you put it on (and will be shown as an override in the RH Engrave panel).