There used to be an add on called: "Copy Attachment To Clipboard" that made this possible. But now it is outdated and abandoned. This is a native function in Outlook and allows one to copy an attachment and paste it in other applications which support copying and pasting files.
Although I think Thunderbird is by far a more complete and efficient client than its Microsoft counterpart, my team just made the switch back to Outlook just for the lack of this feature. Since their workload is enormous and it has to do with transferring such attachments from the mails to our dedicated software, it is understandable.
Hope Mozilla makes this possible in the near future.