BIM 360 is a web platform, whereas a Revit add-in is a .NET assembly executed by Revit.exe on the Windows platform.
Therefore, they are obviously not simply interchangeable.
However, Autodesk has indeed implemented a possibility to access an environment, the Forge platform, that enables you to run your Revit add-ins in a web environment, enabling a tight integration with BIM 360:
https://forge.autodesk.com
So the answer to your questions is:
Yes, you can run your Revit add-in in BIM 360 by converting it to Forge Design for Automation for Revit, or DA4R:
https://forge.autodesk.com/en/docs/design-automation/v3/developers_guide/overview
The Building Coder has discussed some such conversions:
https://thebuildingcoder.typepad.com/blog/about-the-author.html#5.55
Please address all further questions regarding DA4R to the official Forge help channels, not this forum:
https://forge.autodesk.com/en/support/get-help
Best regards,
Jeremy