Drag-and-drop Outlook message into browser
Free plugin solution for older browsers and older Outlook:
https://tonyfederer.github.io/OutlookFileDrag/
From the website:
When you try to drag and drop from Outlook, Outlook correctly identifies the format as virtual files (CFSTR_FILEDESCRIPTORW) since the files do not exist directly on disk. Instead, they are contained in a PST file, OST file, or on an Exchange server.
However, many applications do not support, such as web browers and most .NET/ Java applications.
To work around this issue, Outlook File Drag hooks the Outlook drag and drop process and adds support for physical files (CF_HDROP). When the application asks for the physical files, the files are saved to a temp folder.
I looked at https://outlook2web.com/ and https://www.wilutions.info/ddaddin-download.html
but this free one works just fine and it's open source.
Update regarding newer browsers and Outlook:
After updates to Outlook and the dataTransfer API, drag and drop functionality seems to be available in the latest versions of Chrome and Edge via the "items" property of the "dataTransfer" property of the "drop" event. You then have to iterate the items and check if they are files or not and then use them. However FireFox still seems to be incompatible with Outlook via this API. Not sure about other applications. Here is their bug: https://bugzilla.mozilla.org/show_bug.cgi?id=580928
You can vote for a bug in bugzilla if you have an account by expanding the details section of the page.