I tested drag & drop from eM Client to Opus recently (late September) and found it worked for me.
Make sure neither Opus nor the other program are running elevated, as that can block drag & drop from working between applications:
In an email thread, another user also reported that they had a problem dragging from eM Client to Opus until they went into Logitech GHub and changed their mouse report rate from 500 to 1000 Hz. Note that this is the Report Rate setting, not the DPI setting, and isn't available on all mice/drivers:
That solution was quite strange/unusual but suggests eM Client might be sensitive to mouse and computer speed, possibly due to which windows and other controls the drag & drop crosses on the way from it to Opus.
FWIW, in my testing before then, I already had my mouse set to 1000 Hz.