When many scrum teams are working on the same product, should all of their increments be integrated every sprint?

The answer is: Yes, otherwise Product Owners may not be able to inspect what is done accurately

Even if several people working on one same product together, it is still important for the team to find a way that differentiate the things that they done specifically to the product.

When mistakes occurs, this information would let us identify the specific member that made the mistake during the project rather than putting them all in a same boat.