Is there a way to manage all your styles within ArcGIS Pro? I tried looking under Symbology and Options but cannot seem to find a place to manage my custom styles so that I can reuse them in other projects.
Yes it does, but it's quite clunky compared to the one in ArcMap (in my opinion).
You can see it here in the Project pane, I have right-clicked and created a new Style.
You can then open that style in the Project View, and add new items.
Once your new items are created you can modify to suit, or you can save new items from the symbology pane as in your own screenshot.
It does appear that the Project view will only show one type of symbol at a time (e.g. Point symbols, Text symbols, Line symbols) - you need to change the selection in the Show drop-down to see the other types of symbols.
As for using styles between projects, there is an option to Add Style - you can then browse to your style and add to your project.