Nothing is impossible! But there is no point keeping duplicate content on both places.
If I understand it correctly, you want the same content without manually have to enter it twice on both sites.
Regarding using the same database:
There are two values in the options table: siteurl and home which are used. Using the same options table won't work, even if you update options forcefully for each php run.
So you would need to use two databases.
You can use a mysql replicator, to replicate content from one database to another. Make sure to trigger a script at the end of replication to update the siteurl and home to the other domain values.
You need to ensure that no content anywhere has urls which points to the same domain. If there are any such urls, you need to have procedures to replace the urls after replication.
If your customer need only posts/pages content copies on both places, without the pain of manually copying them over, you can have them post the content via email:
Have both instances read via email and make posts/pages from it.