Auvinen In this case your friend is not subject to the payment of the dogs tax.
There are lots of laws which apply for the whole country. However, many regulations in Switzerland should be studied individually on the cantonal level including dogs tax question.
E.g. your friend as an exempt from payment of the dogs tax must nonetheless pay a fee for the control mark in the canton of Geneva. The control mark is obligatory in Geneva for all dogs. However, this does not exist in the e.g. canton of Vaud.