The banking facility is provided by Australia's 5th largest bank [1], which has been in business since 1858, and is as solid as any bank.
Up Banking provides the user-facing front-end, which of course needs to be highly secure too, but the tech team at Up Banking is one of the best teams in Australia (it caused quite a bit of buzz in the AU developer community when the team was quietly coming together to start working on this a few years ago; As a fun fact, some of the initial dev team previously worked together building The Conversation [2], which is a huge platform and major content source here on HN).
So, they know what they're doing and they know who to bring in to ensure they've done everything right.
Those interested may want to check out this interview with the founder/CEO [3].
My interest: no personal stake or involvement, but am friends and former colleagues with a few of the team members (including the author of this post), and I know of several others by reputation.