Access doesn’t create a new table, it just shows you the same table 2 times. It needs to do that to be able to display the second relationship.
You can confirm that by querying the MSysObjects table, looking for an object called ‘tblPerson_1’. It will not exist.
You can also query the MSysRelationships table, and you will see that the szReferencedObject field is the same for both relationships.
So, don’t be afraid, a new table is not being created.
You’ll probably need to edit the code which Access is using to create the second relationship and give it a different name.
Add another one copy of the table in the diagram and make link with it.