Discord uses battle.net integration and displays player status such as "Playing Overwatch 2". I looked through battle.net API documentation and couldn't find an API that would provide such data.
Is it using some private APIs?
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First, as indicated in the comments, the "Playing [game name]" status is not unique to Overwatch 2. Discord can look at the client side processes, match them against a database & update the reported status accordingly.
So it isn't that Discord is using the various APIs of various games, but that games are able to use the Discord API to make use of Discord's features.