I believe you need to use com drag and drop which works between two applications. Vcl Drag and Drop does not function between applications. The only component that I know of that does this with modern versions of Delphi is Raize DropMaster 2.
DropMaster 2 is a set of 4 native VCL controls for use in Delphi and C++Builder. While the VCL components included with Delphi and C++Builder permit drag and drop between windows in the same application, DropMaster allows developers to add support for drag and drop between applications. The drag and drop can be between the developer's new application and existing applications such as the Microsoft Office suite, a web browser, etc., or between two custom-written applications.
There are several open source components to do this but they have not been updated and will not compile with modern XE versions of Delphi. Melanders Drag And Drop Component was last updated in 2010 so it is 4 years old now and if you try to compile it with XE versions of Delphi there are lots of lines that will not compile. You may find it here: http://melander.dk/delphi/dragdrop/