How to identify word in a sentence representing the song genre?
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.