Why there are cow statues at Shinto shrines
This young woman in kimono touching the head of a statue of a reclining cow at Kitano Tenmangu Shrine in Kyoto is very likely a student praying for good results on her exams.
Tenmangu shrines are dedicated to Tenman-Tenjin (天満天神), the god of
scholarship and learning, who is actually the deification of Sugawara no Michizane, a Heian-period scholar and politician, who died in exile in Kyushu.
Dazaifu, the place he was banished to, also has a big and important shrine commemorating him. Legend has it that when he was sent into exile cows in the area wept, so these cow statues have come to be associated with wisdom and learning.
However, I recently saw a sign at Naruko Tenjin shrine in Tokyo asking people to refrain from touching the statues as a countermeasure against the spread of the corona virus. It is yet another interesting way in which COVID-19 is affecting traditional culture.