How to create a Document Categorization Classifier for different contexts of Documents
This depends on whether you have enough training data for all languages. If yes, doing language ID and language-specific models might be a good choice, especially if there is a BERT-like model available for each language.
An alternative would be, do the language ID and than machine-translate the input into English and only train an English classifier. You can use e..g, high-quality pre-trained Marian models recently published by the University of Helsinki.
Otherwise, I would use pre-trained multilingual representations (probably XLM-R that is much better than Multilingual BERT) to get representation and train a single classifier for all languages. The multilingual representations seem to have even some zero-shot abilities, i.e., the classifiers seem to generalize even for languages that are no in the training data.