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As I found this answer myself while experimenting with Outlook New, I will share my findings.
I posted C# code for parsing the "Chromium Web Custom MIME Data Format" here: https://stackoverflow.com/questions/68745902/what-is-the-data-format-specification-for-chromium-web-custom-mime-data-format/79941017#79941017
For Outlook New, this does not help much: the value is just a JSON string with metadata of the dragged mail.
If you want the real mail (or a mail attachment) as a plain file, you have to implement async drop.
This code sample demonstrates it: https://github.com/Nussbaum-Transportation/new-outlook-async-dragdrop/ (horrible interop code, but it works).
And .NET10 will bring async drop for WinForms applications out of the box: https://github.com/dotnet/winforms/issues/13422 which reduces the code amount for async drop to just a few lines.