Unfortunately that is a new feature coming with V6 (present currently on the nightly builds aka. v5.99 used for testing). for the current stable release it is not possible to copy/paste between different instances.
The new project into which you want to paste is active
Create a hierarchical sheet and go into it
File -> Append Schematic
Copy from there
Go to the upper level and paste
When you’re done, delete the temporary hierarchical sheet.
The workaround above by Meterman seems to work only if eeschema is in standalone mode, i.e. not opened through a project. It also works only if the libraries are common to the two schematics (projects).
Kicad (eeschema) has been started sometime in early 90’s. That history comes with a baggage, and this is part of it. In version 5.99 most of the history has been cleaned and eeschema behaves as a decent modern application.
eelik: You are correct that eeschema has to be run in standalone mode to be able to do this. It was a good and easy way to ensure I got all the inter-board connectors in my project just right.
Once I was ready to start translating the schematics into actual PC board layouts, THEN I switched from standalone mode to individual projects.