I am training a model to identify a word that represents a song genre given a sentence. For example, the model is given a sentence "Beethoven songs are part of the classical genre." The model will categorize the word, "classical" as "song genre". Can I use an RNN classifier to do this or is there another algorithm that can accomplish this task?
You are describing named-entity recognition (NER) which seeks to locate and classify named entities mentioned in text into pre-defined categories. In this case, the pre-defined categories are music genres.
There are many algorithms for NER. In addition to Recurrent Neural Networks (RNN), Hidden Markov Models (HMMs) and CRF (Conditional Random Field) are often used.
In most cases, the algorithm is not the most difficult part of NER. The difficult part of NER is creating a large, accurate annotated training corpus.