For example:
A company that requires all employees to wear light-blue shirts on Fridays.
A government that requires people to file paperwork every time a pet dies.
Bureaucracy
: a large group of people who are involved in running a government but who are not elected : a system of government or business that has many complicated rules and ways of doing things (m-w.com)
Micromanagement
: to try to control or manage all the small parts of (something, such as an activity) in a way that is usually not wanted or that causes problems (m-w.com)
And finally, an allusion that really sums it up: "On Wednesdays, we wear pink."
There is an idiom called Red tape that is closely related:
Red tape is an idiom that refers to excessive regulation or rigid conformity to formal rules that is considered redundant or bureaucratic and hinders or prevents action or decision-making. It is usually applied to governments, corporations, and other large organizations.
Examples and a definition from the book "Idioms in the News - 1,000 Phrases, Real Examples" By Peter Bengelsdorf:
Not exactly what you asked for, since you seem to be specifying a noun, but "Kafkaesque" is a kind of pretentious adjective that's often used to characterize a place or experience as involving lots of alienating, oppressive, and/or otherwise pointless and circumlocutory rules (things that characterize Kafka's works, like the novel The Castle, in which a man is endlessly held back from making a simple entry into a castle by a succession of ridiculous rules and regulations).
If you are ok with American slang: this is a TPS report.
After its use in the comedic 1999 film Office Space, "TPS report" has come to connote pointless, mindless paperwork,[2] and an example of "literacy practices" in the work environment that are "meaningless exercises imposed upon employees by an inept and uncaring management" and "relentlessly mundane and enervating".[3] According to the film's writer and director Mike Judge, the acronym stood for "Test Program Set" in the movie