Did the Companions, may God be pleased with them, sway or shake when remembering God, and if they did not do that, is it permissible for us to shake when remembering God?

The mentioned narration about the Companions shaking upon the ذكر (remembrance) of Allah has not been authentically reported. What resembles it is found in a weak narration attributed to Ali (may Allah be pleased with him). Shaking, dancing, or swaying upon the ذكر of Allah Almighty with the intention of worship is not permissible; it is an innovation (bid'ah).