Whenever I want to adapt the issue tracking list to get something else, I can delete all fields except category (which does not have the button DELETE), Is there an explanation?
There as some default columns created by SharePoint based on content type, Category column belongs Content Type "Issue", Other than category some more columns like Title, created by, modified by, created, modified also cannot be deleted.
Though I could not think of any other logical reason to not allow us to delete this column, this is very well possible that we don't want to use category for our issue list.
One alternate to hide(not delete) is to "modify content type", first allow management of content type for issue list, edit content type and select Category column. you will see one option "Hidden (Will not appear in forms)" in column settings.
May be this would help to hide this column from Issue list.