What is special about Preview added text or shape elements? Can I move other elements besides these?
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…).