People get all sorts of messages, and it doesn't seem to affect anything about the second app.

What is the second app for? Do either of these offers have a "first-time only" language? Or is the second card of the same type (Thank You Premier) as the first card? If neither of those, there's no big need to apply two-at-a-time. The general Citi credit limit rule is no more than two card apps (successful or not) every 65ish days. But that doesn't mean they have to both be at the same time. One (non-AA) card app every 35ish days is the same to Citi as two cards apps at the same time every 65ish days. They only care that you wait 65ish days from the first card before you apply for the third card; they don't care in general when you apply for the second card.

The cases where they might care is where the two cards are the exact same type, and yet you still want two of them. And then the other case is AA, which has its own special rules (on top of the general rule above).