If you get any variation on desktop picture, your Mac I working as designed. But there are MANY options, so it may not be working as you intended. It sounds like you have invoked "Extended desktop", where the second display is concatenated along an edge to the built-in display, forming a larger extended desktop.
System preferences > Displays > arrangement ...
...will show you how your Mac thinks the displays are arranged, and you can change their placement to reflect the physical displays on your bench by dragging. then the mouse will move seamlessly across the boundary, and can drag a window all the way or part way across onto the other display.
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