Currently, it appears that spatial bookmarks in ArcMap and Pro as created with the "Bookmarks" options, do not store the Coordinate System and Transformation of the Data Frame in ArcMap, or Map in Pro.
As a consequence, when switching the map to another coordinate system, and subsequently selecting a bookmark to return to a previously visited extent, the zooming or panning will not reflect the actual extent and Coordinate System of the originally bookmarked location. You need to manually reset the Coordinate System to be able to re-visit the exact same extent as stored with the bookmark.
It would be really desirable to see the spatial bookmarks store the Coordinate System and Transformation as well, so that zoom or pan option truly return you to the exact extent of a region bookmarked by automatically resetting the Coordinate System and Transformation of the Data Frame or Map in Pro, instead of taking you to a rough approximate location of the stored bookmark.
This is especially useful for accurate multiscale map documents, where the user may have different layers for different scale ranges in a single map, that may be best viewed in specific coordinate systems tailored to visibility scale range of the layers (e.g. Continental / Country level).
Related to this Idea although not specifically mentioning Coordinate Systems: