A number in parenthesis is appended to my Apple TV's name when shown in the AirPlay menu

I named my Apple TV (4th generation) "Kirsch TV". After a few hours, in the AirPlay menu of my iPhone it was called "Kirsch TV (8)". Today it is "Kirsch TV (333)". For reference, it is connected to an AirPort Extreme 802.11n (3rd Generation). See screenshot:

I used to see this bug a lot on one of my Macs, back when I had (I think) Yosemite installed. I know it has to do with Bonjour naming collisions and the Airport Extreme doing some of the Bonjour service announcing for the Apple TV when it goes to sleep. But I don't know how to keep it from happening.

Help is appreciated.