The actual answer is that the starting distances are nonzero, and with nonzero starting distances, you can get any size with the right rate of expansion. The "initial singularity" with infinitely small size is a mathematical anomaly, a result of naively running things back in time further than we know how to.

No rate of expansion, when multiplied by 0, gives you a positive distance. Something with infinite density cannot expand to something with finite density. The initial state of the universe as far as we are able to talk about it is hot, dense, and low-entropy, but it is not infinitely small.

So, any answer that involves some rate of expansion as the reason is incomplete, because without nonzero initial distances, you just get zero expansion.

Now, with all that out of the way, the answer to "how can the universe be 93 billion lightyears across if light has only had 13.7 billion years to travel" is just that the way we talk about the size of the observable universe is about how far apart those things would be from us now, at this moment in cosmic time (their comoving/proper distance). The majority of the observable universe is already unobservable, if by "unobervable" you mean "light emitted from it right now will never reach us". But we can see their past, and thus they're part of the "observable universe".