"Shinkan" is a person serving a certain God or working as a government official at a facility where God is worshipped. There are no Shinkan in Japan at present.
As for Shinto religion in Japan, old materials show several cases in which shinkan was used as a term designating a Shinto priest (a person involved in religious services and office work in a shrine), and it is still used as common name for Shinto priests in everyday talk. When Jingikan (the department of worship) was reestablished in the early Meiji period, all Shinto priests became government officials (Shinto priest is equal to Jinkan,) but this was only applied to the Shinto priests of Ise-jingu Shrine and Kankoku Heisha (a general term for Kanpeisha and Kokuheisha that were high rank shrines) due to the system's problems, and since 1894, only Shinto priests in Ise-jingu Shrine have been treated as Jinkan serving as government officials. After the Greater East Asian War (World War II) was lost, the Jinkan related system was discontinued based on the Shinto Directive issued by the General Headquarters of the Allied Powers.
In ancient Egypt, Greece, and Rome, people (mainly men) who served at shrines were called Jinkan.
Subculture and explanation of Shinkan
Shinkan appearing in fantasies and roleplaying games means a clergyman who is not related to a particular religion, or a clergyman in Christianity (a priest or pastor) can be called Shinkan.