[If you mind spoilers, please scroll away]
"Actually, it doesn't matter. It really doesn't. You guys can do whatever you want."
– Monika (Act 2 Chapter 4)
I could stop there with just that comment because it is most likely the actual answer. However, because I am bored, I'll be discussing this topic. If we are not taking Monika's word for the entirety of the game (which makes this entire analysis invalid and pointless), then I believe this question has now been solved. To put it simply, Monika is a 3rd-year student while the rest of the characters are 2nd-year students.
Let me also get this out of the way: there is no evidence or hint that determines if a character has repeated a year or not, meaning everyone except Monika could have repeated their first-year curriculum and Monika is actually a 2nd-year student at the time the Literature Club was made. It could also mean the protagonist is the one to repeat while everyone else moves on to be 2nd-year students and Monika be a 3rd-year student. For the sake of the discussion, it's going to be assumed no one had repeated and the characters made progress in school the entire time they are there.
The critical piece of evidence is the real-life secondary education system in Japan. Students would have footwear known as "*uwabakis*," and they would be coloured to represent a student's differing grade among another (according to various websites from googling).
Monika is the only character to have uwabakis with pink tips in contrast to the rest of the cast's light-blue tips. The protagonist's uwabaki colour is unconfirmed because of the lack of objective evidence, but for the purpose of the evaluation, let's assume his uwabakis are light-blue as well since he did not repeat his first-year curriculum.
The colour difference means Monika is a year below or above the rest of the cast. However, the protagonist mentions Monika being in his class "last year," so shouldn't he and she be in the same grade? According to Heading 3 Section 2 on this site showcasing Japan Education facts for **elementary and junior high schools** (meaning this can be false in senior high schools; please confirm), a student can be "admitted" up or down a year if they prove to be an exceptional/outstanding student during their semester and can continue working with their 1st-year classmates until the time comes they have to go to their appropriate classroom.
This means it is possible that Monika moved up a year during the time she was a 1st-year student, and she either moved to the 2nd-year class or stayed in the 1st-year class with the protagonist. This also means that by skipping a year later to the time the Literature Club was made, the protagonist would be a 2nd-year student, and Monika would be a third-year student.
A question pops up, however: how do we know Monika went up a year? She could have gone down. The answer to that question is in Monika's character and the logical sequence of scenarios afterwards. By all definitions, Monika is a Mary Sue, a character that is seemingly perfect until it is revealed later their tragic backstory. It would go against her character and goal if she were to be downgraded a year since it implies flaws. Regarding the scenarios afterwards, If Monika went down a year, she would likely have not been considered for leadership in the debate club because she could have repeated for failing tests, being anti-social, not be willing to participate in school, and so on, each being opposite to how Monika is looked up to by other characters and how she presents herself.
Considering the evidence apparent to us in the real world and using that to explain and guess the grades of the characters, it can be concluded that Monika is definitely a year above the other characters and realistically be a 3rd-year student and the others 2nd-year students. Monika wants to be the most appealing girl, so she would not give herself flaws or reveal being downgraded a year as a truth, rather she would be seen by others as having such quality as a student, she would move a year upward and be given various opportunities. Therefore, her uwabaki tips being a different colour than everyone else's is justified by real-world policies and logic.
Let me know what you think. Did I miss anything? Got a question? Am I wrong about something? Let me know. In the meantime, I hope this was an interesting little theory to think about.