I suspect that your document is either one of the following:
(a) A scanned PDF document containing no actual text
Can you tap over text and see if you can select it? If you cannot, your PDF document is a scanned PDF.
(b) A secured/encrypted PDF document that does not allow any changes including adding comments
Do you see the lock icon in the bottom bar?
When you try to switch to Comment in the Tool Switcher (see the screenshot below), do you see an alert message "This tool is not available due to the document's security settings."?
Because of the document security & restriction settings, your PDF document is encrypted.
Acrobat Reader mobile products (for iOS, Android, Windows Phone) have the known limitation where changes cannot be saved to encrypted PDF documents. Therefore, users are not able to make any changes (e.g. adding comments/annotations or filling out forms) to encrypted PDF documents.
As you already know, there is no such limitation in the desktop version of Adobe Reader or Acrobat.
If you are the author of the PDF document or your know the password, you can remove the document restriction settings so that the PDF document will no longer be encrypted. But you need to use the desktop version of Adobe Reader or Acrobat Pro/Standard to change the settings.
Sorry for the inconvenience.