Most people do not intuitively think that shrines are related to trademarks.
However, some shrines own trademarks in Japan.
Looking at the website of Yushima Tenmangu which is one of the famous Japanese shrines, we can find the statement like "we are the owner of registered trademarks of 湯島天満宮（Yushima Tenmangu）".
All over Japan, each shrine sells small charm (omamori) or gifts such as stationery with shrine name. In that sense, trademark registration is necessary.
Regarding Yushima Tenmangu, it owns more than 50 registered trademarks designating wide range of goods or services in Japan. We do not know whether it is practically effective to designate widely, for example, designate electronic machines or furniture.