For example, if you say you’ll be there at 10, you better show up at exactly 10, not 10:04. My son is like this. If we make a rule, we cannot change it without a fight.
I would describe such a person as fastidious
giving too much attention to small details and wanting everything to be correct and perfect
(of a person or their behavior) unlikely to be swayed; resolute. Lexico
You describe someone as unyielding when they have very strong, fixed ideas about something and are unlikely to change their mind. Collins
Miss Kuhlman said that initially waiting for their turn was rather chaotic, but after the first couple of weeks, they understood that waiting politely for their turn was the only way they would be able to ride the tricycle. Miss Kuhlman was unyielding about her rules. B. E. Lithander; Life's Pages—A Story Worth Sharing (2010)
He thought as a judge down to the time when he had made up his mind, and after that his obstinacy was unyielding. The American Law Review, Vol. 25 p.25 (1891)