Copy a geometry from one feature to another without attributes?

Yes it is possible, see instructions below:

1. Start an edit session on both layers that you want to do the copy/paste between.
2. Use the Select feature by rectangle tool to select the feature in the map (copy from).
3. Click the Edit (menu)>Copy Features button.
4. Highlight the layer in the TOC that the feature will be pasted to.
5. Click the Edit (menu)>Paste Features button.

For copying just the geometry between existing features there is the plugin Geometry Paster: https://github.com/nextgis/geometry-paster