ESA neither launches or builds rockets. They are an end-user, and they own the launch infrastructure in Kourou. Both these roles will exist long after Ariane is gone.

Arianespace does not build rockets. It's a launch services provider, they purchase the rockets from other companies, handle all the logistics and manage the launchpad. It's in no way married to Ariane 6, and it has in the past and still does sell launches on multiple different rockets from different companies. There will be demand for these services even after Ariane the rocket is gone.

ArianeGroup is the company that builds Ariane 6. They are a complex co-owned project between Airbus and Safran, with workshare distributed between France and Germany. They make other things than the Ariane 6, notably the French SLBM. Eventually the Ariane 6 will go away, because it will be obsolete, but that will not necessarily mean that ArianeGroup will go under, because they have an another subsidiary called ArianeWorks developing a new rocket, called Themis.