There is no way to import documents from cloud storage such as Google Drive or Dropbox directly from the Acrobat app. You can open documents from the cloud and at first you might think this is sufficient but you lose the function of beginning on the page you were last on when you close and reopen the app. The only way to import documents into the app is by going to the source location and doing the "Copy to.." function. The ability to open documents from the cloud and not be able to import directly from the Acrobat app and the loss of metadata such as page location is cause for confusion. There should be a function after opening a document from the cloud to import it locally.
If you want to make a separate local copy of a PDF document by importing it from cloud storage into Acrobat Reader on iOS, you can use "Copy to" (or previously known as "Open in"), as you have already pointed out. However, any changes (e.g. filling out a form or add annotations/comments) to the locally stored PDF document will not be saved back to the cloud storage automatically.
Acrobat Reader for iPhone and iPad provides Dropbox integration and Document Picker support to open documents directly from cloud storage.
Please take a look at the following section of Acrobat Reader Help to see if it fit your needs.