This problem has been fixed in ArcGIS Pro 2.5.
By default, focus in the browse dialog box is in the Name text box, even if clicking a disk on the left side of the dialog box. The expected behavior is observed when using the Shift-Tab key combination to put the focus on the contents list instead.
If the Name text box has focus, typing type a letter into the dialog box when the focus is there, an item is not selected in the contents list. However, the Name text box auto-completes what is typed using items that are in the contents list. Select an item from the auto-complete drop-down list, and then press OK or Enter, and the indicated folder (in this case) is either selected or browsed into, depending on which browse dialog box being using. In the Open Project dialog box as indicated, if the item identified in is a folder, the browse dialog box then directly browses into the folder - the contents list does not shift to show this folder as being selected first, the contents of that folder are seen.