> What is their case then exactly?
Speculating, but TOS generally say that you're not allowed to share your password with anyone. Assuming that's true for the AA site, AA might argue that TPG is encouraging users to break their TOS ("tortious interference").
Though if app doesn't actually share the password with TPG and just uses it locally there may well be a question of whether entering your password into a third-party app actually counts as sharing it with that party. How exactly would this be different logging in from a web browser? It's just a different kind of user agent. Are Google, Mozilla, Microsoft, and Apple guilty of tortious interference simply because the software they release has access to your passwords on your own machine and could report them back? (For that matter, they even store the synced passwords on their servers, though in principle those are supposed to be private.)
Of course there could also be specific terms in the TOS against accessing the service with unapproved user agents, independent of any prohibition on sharing credentials.