What is special about Preview added text or shape elements? Can I move other elements besides these? I found that if I use Preview to open a PDF file, and use its tools to add text, or shape elements such as arrows, ellipse, rectangle, then I can use Preview to edit them.

But if the items are not Preview's, then I can't edit them.  In this case, I have also tried using Adobe's Acrobat Standard or Pro trial version to edit a file, and when I see a person wearing a tie, with a shadow, I can move the tie to the left and move the shadow to the right (not always, for example, if the person with the tie is a pure image, then I can't).

So, can Preview have this capability?  Why can I move element that Preview added to the PDF but not move other elements in the PDF?
The items you add using Preview.app (text, shape elements, etc.) are annotations (technically correct term), also called "comments". 

The example you mention to move around using Acrobat are part of the contents of the document. 

Annotations and contents are two completely different things in a PDF. 

Acrobat (Standard, Pro) has the tools to access both. Preview.app (and Acrobat Reader) have the tools to access annotations, but not to access contents. (a bit of a rant: …and that's good so, considering how badly Preview.app can mess up PDFs…).