Answers to the question Why are passports checked at the check-in counter, Immigration and even at the gate? point to the fact (among others, but this is the main trend) that airlines are held responsible by the arriving country if they did not do some due-diligence on the capacity of an individual to enter the destination country.
The Schengen Agreement allows, in normal times, for free travel between Schengen countries. I therefore cannot think of a reason for the airlines to check the IDs or passports of the travelers at the gate.
I remember that 15 years ago the check was random (sometimes they would check for the ID, sometimes not) and further back there was no check at all.
Since you can drive between EU countries freely the "can they go to the other country" aspect is moot.
The ticket is already payed off so whoever travels does not make a difference here (except if there are fees for changing the name (which is the case at least for low-cost airlines), but this is not the problem of the airport and their staff)
Is this just tradition or security theater, or is there a specific EU reason for that?