[D] Is there a formal name for "dialogue classification?"

I'm trying to classify dialogues into categories. Specifically, if we have customer service chatting data where clients ask questions and CS agents answer, I want to categorize those to have labels like "product inquiry," "delivery inquiry," etc.

Is there a formal name for this? It doesn't seem to be normal text classification because we have to take into account speaker information. There also doesn't seem to be a task called dialogue classification. I thought intent classification may come closest but …

agents classification customer customer service data delivery dialogue etc labels machinelearning normal product questions service text