I think you're unnecessarily hyperventilating over this. My guess is
that someone either provided or copied a wrong tail number at the
FBO. When they discovered the bill hadn't been paid, they did what
FBOs usually do in this situation - look up the FAA info and send a
bill.

All you have to do is discuss it with them. I had a similar situation
a number of years ago and just wrote them back that my Tripacer would
have considerable difficulty carrying 75 gallons of fuel, and that I
had never in my life visited their facility.