It is a tradition that the incoming first lady hands a present to the outgoing first lady on Inauguration Day. There’s a 2001 video of Michelle Obama giving a box present to Laura Bush in the same spot in the same situation.
Laura Bush simply held the box on her side and smiled during the photo op.
s/2001/2009/
The report I saw said it was a faux pas by Michelle to hand a gift to Laura Bush, and a repeated one now, not a tradition, not something done before Michelle. I don’t know if the report was correct.