Is a random intercept multilevel model the same as a fixed effects model? Let's say I have the study time and test performance of a group of students who are clustered as classes.
Is using a multilevel model (student, class) the same as using a fixed-effects model, where the classes are treated as dummy variables? Based on my coarse understanding, using a multilevel model means estimating a fixed coefficient but letting the intercept vary according to the classes. This sounds exactly the same as treating the classes as dummy variables so that the intercept is let to vary.
Am I missing something? What is something that only the multilevel model can do that the fixed effects model cannot?
 A: Initial Response
Let's say you have 5 classes represented in the data you use to fit each of your two models.
The fixed effects model assumes that the 5 classes are the only classes you are interested in and that they are unrelated to each other (i.e., knowing something about the relationship between study time and test performance for the students in one class will tell you nothing about that relationship in another class). The fixed effects model will therefore estimate the relationship between study time and test performance for each class using only information from that class.  The model will enable you to compare this relationship across any two of the 5 classes.
The multilevel model assumes that the 5 classes are a representative sample of a larger set of classes and that the classes in this larger set are similar to each other in terms of the relationship between study time and test performance.
In particular, this implies that there is an "average" class in this larger set and that all other classes in the set are either above or below average in terms of the relationship of interest. In other words, the classes in the larger set are more alike to than different from the "average" class. Put another way, the multilevel model assumes the classes "are not isolated and potentially have some commonality", as explained in https://m-clark.github.io/posts/2019-05-14-shrinkage-in-mixed-models/.
The multilevel model will estimate the relationship between study time and test performance for each class represented in the data used to fit the model by using information from that class and from all the other classes similar to it which are included in the data. This "borrowing" of information for estimating relationships is something specific to the multilevel model which you will not encounter in the fixed effects model.
After fitting a multilevel model, one typically describes what the relationship of interest looks like for the "average" class and also the extent to which the other classes in the larger set of interest differ from the "average" class with respect to this relationship.  There is no longer an interest in describing how each of the 5 classes represented in the data differ from each other with respect to the relationship of interest since we don't care about the 5 classes - we just care about the larger set of classes they represent.
Additional Points
1. Number of Observations per Class

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*In the fixed effects model, a sufficiently large number of observations per class should be available for a satisfactorily accurate estimation of the fixed class effects.

*In the multilevel model, the number of observations per class can be small; even one observation per class may be enough for a satisfactorily accurate estimation of the random class effects, provided some classes have several observations.

2. Class-Specific Predictors

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*In the fixed-effects model, class-specific predictors cannot be used as additional predictors in the model because their use would lead to an unidentifiable model.  (Examples of class-specific predictor would be class size, with possible values "small", "medium", "large", or *percentage of male students in the class.)


*In the multilevel model, class-specific predictors can be included in the model and used to model the between-class variability with respect to the relationship of interest between study time and test performance.
3. Sources of Variability

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*In the fixed effects model, there is only one source of random variability, whereas in the multilevel model there is more than one source of random variability. See https://www.theanalysisfactor.com/understanding-random-effects-in-mixed-models/.

4. Prediction for New Classes

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*The fixed-effect model does not allow prediction of the outcome variable (i.e., test performance) for (students in) classes that were not included in the data used to fit the model, but the multilevel model allows this type of prediction.

