No, and not by policy. By design.
Apple's blocking system hands apps like Aura a set of opaque tokens rather than app names. Those tokens only mean something on your own device. When Aura applies a block, it passes tokens back to iOS, which does the actual restricting.
So there's no list of your apps on our servers, because Apple never gives us one. The same is true of every app built on this system.
Your screen time statistics are handled the same way, on your device.