I am trying to understand if the F1 scores are higher for a binary classification problem than for a multiclass classification problem.
In general the lower the number of classes the easier it is for a classifier to assign the right category. However this completely depends on the characteristics of the data, in particular how well the features match the classes.
For example if one tries to somehow classify pictures of dogs, cats and rabbits into two classes, it's possible that the performance will be poorer than into 3 classes. This is an obvious example but sometimes the data might contain patterns which fit more easily into 3 groups than 2.