Will a dataset with multiple labels perform better than with binary labels? Suppose I have a dataset comprised of garbages. Will a model perform better if I only label the dataset with biodegradable or non-biodegradable?
Or will it be better if I label them with plastics, paper, cardboard, glass, and organic?
In fact, I also plan to further increase the number of labels for example, the plastic label with be comprised of a large number of brands of plastic wrappers, etc.

I think that having a large number of labels is detrimental to computational performance in both training and evaluation since for example, Linear Discriminant Analysis will not lead to much reduced dimensionality due to the nature of the subspace spanned by the centroids on each label.
Neural networks would have a very wide top softmax layer and I am pretty sure that would require a wider architecture.
Aside from model issues, I could easily suffer from class imbalances. Is there any merit to having multiple labels ?

Generally speaking, having classes that are not naturally ordered creates a more difficult prediction/probability estimation problem.  When a binary problem turns into a multinomial/polytomous problem the task is more difficult as more model parameters exist, the effective sample size decreases, and the sample size required to achieve a given predictive reliability increases.
If some of the classes are connected, i.e., types of plastic, you can gain information and predictive reliability by using a Bayesian model that has prior distributions for parameters such that you borrow information (assume some similarity of effects) across connected categories.  This reduces the effective number of parameters.  Sometimes this can be done with Bayesian hierarchical random effects models.
When you can assume that all levels of the outcome variable are ordered, everything changes.  The effective sample size increases and the number of parameters that matter does not increase with increasing number of outcome categories.  Models such as the proportional odds ordinal logistic model come into play.
