The Senate has held precedent that there is no basis in impeaching members of the Senate (as they argued that they are not 'officers of the United States), and they have their own process for removing someone from office, which is voting to expel a member of the Senate with 67 Senators consenting to the expulsion, and this goes back around 200 years to the expulsion of William Blount.

Now, Paul is very free to try and do the exact same thing to Schumer, but I just wouldn't expect him to get very far.