If you have 6 tables that would use exactly the same layout, then there is a very real possibility that you should only have one table but with a "category" field that lets you work with 6 different groups of records on the same layout--which would eliminate the need for what you are describing.
You can enter layout mode and select Duplicate Layout from the layout menu. Then enter Layout Setup... to select a different table occurrence in the Show Records From drop down found there. But if you do that, you then have to double click each field on your layout and reselect the field in Specify Fields to come from the newly specified table occurrence. If you have any portals on the layout, they need the same update process including an update of "Show Related Records From" in Portal setup.
It's also possible to copy and paste all the objects on a layout and with a bit of "sleight of hand", you can avoid having to respecify fields: Importing Layouts