There are several solutions to manage incoming Instagram messages. It all depends on the volume of chats your team faces and the number of messages received daily. This will then affect the solution to be implemented to manage all Instagram messages: remember that managing 100 messages a day is different from managing 1000.
To date, Instagram has only provided the mobile application to manage user messages. However, a few days ago access to the API was enabled to use Instagram Direct with external tools such as Callbell and thus be able to manage a customer service or a sales team regardless of the volume of messages and requests received.