Call routing using AI I'm trying to find out how AI can help with efficient customer service, in fact call routing to the right agent. My usecase is given context of a query from a customer and agents' expertise, how can we do the matching?

Generally, how is this problem solved? What sub-topic within AI is suitable for this problems? Classification, recommender systems, ...? Any pointers to open-source projects would be very helpful.
I would recommend to have a look at intent classification. Once intent of the conversation is known, it could be routed to specific departments.

Basically in chat bot development, we try to do intent classification and entity extraction.

Intent classification is simple text classification and you would need training data.