Keep Unity's movement, rotation, and scale interface

While in the scene editing mode, I can select an object and move, rotate, or scale it, using the mouse.

While in play mode, during run-time, I want the user to be able to edit the scene using the mouse, in the same way. Is there a way I can keep this interface and functionality in play mode?

I am building it as a standalone application on Mac.

No, You can’t include Unity editor into a built application. You have to create your own set of tools for that purpose.