Without claiming any knowledge of this particular case, the FDA has processes for accelerating approval for medication for which mass efficacy testing can only be done in the long-term, but testing for adverse side-effects is already complete. The recent approval of Alzheimer's medication[0] is one example of that.

From a quick Google, it seems that Phase III trials are about comparing a new treatment to existing alternatives, rather than monitoring side-effects. Perhaps the testing simply can't be complete before 2023, since it involves checking long-term immunity? In this case, the FDA may have approved the vaccine because they are confident that the risk of negative side-effects is sufficiently small, and that the vaccine gives some benefit - even if testing hasn't established exactly how much.

[0]: https://www.theguardian.com/commentisfree/2021/jun/28/alzhei...