In an announcement made by Jessica Karle, the Lead Product Designer from Paperless Post when redesigning the online experience she states "We are a group of designers from all disciplines—fine art, graphic design, product design, UX/UI, and branding."
Also, looking at her work she designed some lettering logos. Taking that and adding it to the statement above and the fact that we can't identify the font anywhere I think we can assume that the font was designed especially by/for them.
It's not that common but if you think about the team they have, how long do you think it would take them during a brainstorming for one of them to say "Wouldn't it be great if we took the time and designed our own font?".
Also, searching on Justia Trademarks for Paperless Post registration you will get the PAPERLESS POST - Trademark Details which at the Mark Drawing section has 3000 - Illustration: Drawing or design which also includes word(s)/ letter(s)/number(s) Typeset