TL;DR: No, real-life student councils do not have absolute powers and aren't as powerful as depicted in anime or manga.
Japanese school life tends to be much more hierarchical and organized, with students being put in charge of far more of how things operate and thus the concept of student council comes. So first, let's define student council in Japanese schools. According to Wikipedia:
[...] student councils were added to Japanese schools after World War
II. In Japanese schools, students in a class stay together as a
cohesive set in the same homeroom for most of the day. Each class has
one or more elected representatives who reports to student council.
The student council consists of members who are elected by the student
body. The council is often responsible for organizing events such as
the culture festival, sports day, and class field trips. The council
also oversees the school clubs, and has absolute influence on school
and club policies.
Note that it says it can influence school policies and rules but that doesn't mean that they have the absolute power to set and govern school rules. Those are done by schools administrations. To be more precise1:
In anime no one is cooler or more highly revered in high school than the student council. They are are held at a higher prestige, respect, and hold actual authority over other students. While in reality this is just not true at all. The student council does not have authority to make decisions when it comes to school rules or how the school operates. The school administration is held at higher regards than the student council, and age is just as important in regards to authority. One of the pole participants who was part of the student council in high school chimed in on the situation, “I was a member of the student council, and we didn’t have much actual authority.”
So, despite student council being depicted as the absolute power-wielding entities who can revolutionize the world, in real-life they have, more or less, very little power. But don't get me wrong! They are still very strict and disciplined, and at least hold powers to control club activities/policies and set all sorts of classroom rules. In some schools, they are given the power to set and run events like field day or cultural festival and organize class trips. Please read ref. 3 and ref. 4 for an exhaustive discussion.
- Japanese Students Reveal Differences Between Anime High school And Real Life
- 4 ways anime high schools differ to real-life Japanese high schools
- What Are Japanese Student Councils REALLY Like? (Here, it points out how Japanese student council works differently from American schools. It also notes that Only Yesterday and His and her Circumstances have depicted student council accurately)
- Japanese Student Council: Differences between Anime and Real Life (You can find an exhaustive explanation, discussing every differences by pointing out every depiction shown in anime and differentiating them from real life portrayal).