It seems to me that you are using a device running Android KitKat? Is that the case? Android KitKat has placed some restrictions on third party applications from modifying the content. Please read this for more details: KitKat and SD cards — what's fixed, what's broken and what's misunderstood | Android Central.
In the previous versions of the application, a local copy was saved whenever any edits( e.g. adding comments such underline, highlight, freeform, note etc) where made. In the current version of Acrobat DC, if you open a file from the floating action button, it prompts you to make a local copy first so that if any edits are made to the document, they can be saved back. If this behaviour is not acceptable to you and you would like to have the earlier behaviour, you can open the file directly from the SD card using a third party file manager application rather than from the Floating action button. In this case, you will not be asked to create a copy while opening the file but will be prompted for the same if you make any edits to the document.
Hope this helps.