In general it is not possible to use Bridge to embed sidecar XMP information into files like Lightroom is doing it. And Bridge only supports metadata read/write in those formats and in those ways as are defined in the XMP specification.
But you can use the latest XMP SDK to write your own handlers for file formats that are currently not supported. Those file handlers can then be used in either your own application that uses the XMP SDK or any Adobe Video application like Premiere or After Effects to read/write sidecar files or embed XMP metadata into the files. Currently it is not possible to use those file handler plug-ins in other Adobe image applications like Bridge or Photoshop.
Please see the XMPFiles Custom File-Handler Plug-in SDK for more information.
Please note that other applications (like CQ) might probably not automatically pick up the metadata you wrote into the file or in a sidecar file, as the other application might probably not have the "knowledge" or technical means to handle metadata in such file formats where you wrote the metadata in a custom way.