Yes. The dog licence fee is an obligatory local tax for dog owners, which levies a tax for keeping dogs. The amount varies depending on the community or municipality. Under certain circumstances, Leipzig-Passport holders, i.e. people who receive social benefits or support from the state, may be exempt from the dog licence fee for the first animal. At any rate, a significant reduction in fees can be expected.
All the necessary information and forms can be found on the City of Leipzig website (unfortunately, only in German).
You will find help with the application in the Animals Telegram chat or here.