But cities are corporations, they aren't allowed to establish courts like this

This depends on the laws of the state in which a city is located. In Texas, for instance, cities are "municipal corporations" whose only commonality with a "business corporation" is the word "corporation." State law, consistent with the state constitution, gives certain cities the authority to establish a variety of courts and "administrative tribunals" by passing a city ordinance. So, they are very much "allowed to establish courts like this," though the fairness in practicality may be in question.