Quittancer does not mean to give someone a receipt:
Quittancer, verbe trans.,dr. civil et comm. Donner quittance de quelque chose (dette, obligation). Quittancer un contrat, une dette, une obligation. Quittancer un mémoire d'ouvrages faits, de marchandises fournies (Ac.). Au passif. Ce mémoire a été quittancé par le tribunal (Besch.1845)
https://www.cnrtl.fr/definition/quittancer Some sort of documentation or receipt may serve as proof that someone has been "quittancé," sure. But the word means releasing someone from a debt or obligation. A receipt isn't necessary, and may not be sufficient, to do that. The receipt is
evidence of the quittance, in both FR and EN. See above for the FR, or Merriam-Webster for US EN:
"Definition of quittance
1a : discharge from a debt or an obligation
b : a document evidencing quittance"
https://www.merriam-webster.com/dictionary/quittance When people use "quittance" to refer to the piece of paper (receipt etc.), it's a type of metonymy:
https://en.wikipedia.org/wiki/Metonymy#Examples