With QGIS it's easy to add a new part to an existing feature/geometry but I can't see how to copy/paste this part from another existing feature/geometry.
Is it possible?
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Yes it is possible, see instructions below:
For copying just the geometry between existing features there is the plugin Geometry Paster: https://github.com/nextgis/geometry-paster
Old discussion but another answer if your source geometry is in a layer well supported by the DB Manager (PostGIS, GeoPackage, etc.):
SELECT <geom> FROM <layer> WHERE id = <#>
You have the geometry in the original projection without attributes in a scratch layer.