*** SOLVED ***

Just found out that the solution is to make sure that the object copied is already snapped to grid. The snap position (relative to the grid) will then be copied to the paste

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How do I enable snapping upon pasting an object? I am making textures pasting in different objects until I have created a texture. All the objects need to be snapped to grid by bounding box center, but since no snapping occurs when pasting I have to manually move each object after pasting to make it snap. Of course I'd like it to happen automatically when pasting. Is this possible?

EDIT: When I paste the object Inkscape displays a context message "multiple of grid spacing". What does that mean? I'd like it to be changed to "Bounding box midpoint to grid intersection".

Two thoughts, might help.

Ctrl+Alt+V pastes in position.
If you want to position many objects by their midpoint, center aligning can be an alternative
-paste one object, group it, enter group, paste other objects, select all objects in group, center align, exit group.

Thank you Lazur. Ctrl-Alt-V pastes the new object at the exact same location as the old one, so same x,y. Just like Alt-D does when cloning. (is that what "in position" means?).

About center aligning I tried following the steps you mention exactly but it just places all objects at the same x (using center vetical axis) and same y (using center horizontal axis).