I’m not familiar with the system, but I don’t think that’s what the linked page says.
For me, there are two situations:
You don’t use your own tag, and the rental company is on the given list (Apex, Avis, Budget, Europcar, Goldcar, or Sixt).
In this case, it looks like the car already has a tag (or they just use the license plate), and you will get a new Rental account linked to you and that car to handle the tolls. The tolls and the daily service fee (only on days there are any tolls) will be charged to the card provided to the rental company.
Or you can use your own tag. In this case the system will apparently detect the two tags (or the tag and the plate) and use only yours and charge it as usual.
As you won’t have any tolls on the rental company’s tag, you won’t get a Rental account and won’t be charged the daily fee. This is the part that is not the most explicit in their description but I think that’s what they mean.
They specifically advise against adding the rental car details to your account, they somehow know it’s a rental and you can use your tag in it (either because they have a list of the cars of the participating companies, or because they detect their tag at the same time as yours). If you add your license plate you may start getting charges for other users of the car before or after you.
If you rent from a different company then I have no idea what happens. Obviously the first option is off the books, not sure how things should work out for the second one.
You may contact the rental company for additional details.