Can I get a TotalEnergies Aviation card without specifying the aircraft's registration number?

Yes, if you want an Avgas 100 LL, UL91 or JET A-1 with TICPE card. Your card and your bills will be stamped TS APP, for Tous Appareils (all aircraft), instead of carrying the aircraft registration number. TICPE-exempt JET A-1 cards are not available without an aircraft registration number, however, because they have to show the registration number that appears on the TICPE exemption document issued by the customs authorities.