In order for your Pro map to be linked to the Web Map, you need to find the web map in your Catalog, then choose "add and open". In your project's Maps folder, a linked webmap should have a little chain icon.
When you initially share a map to AGOL, the original source map doesn't automatically become linked to its shared counterpart, so Pro won't know that you want to update that particular web map.
- Josh Carlson
Kendall County GIS