The vastly bit is kind of an understatement.
Each stable isotope is indistinguishable from every other carbon isotope. So, you can probably encode a few bits per atom assuming you can somehow read this data. However they are talking vastly larger numbers of digits here. Where there are only ~10^80 digits worth of atoms in the visible universe, but hey bump that to 10^90 it does not help.
Sure you might encode 10^6 or hell I will give you 10^100 bits of data per atom or something but that’s not even close to helpful. It’s still on the order of K atoms in the universe and you want to encode 10^400+ * K bits of data. So each atom needs to encode 10^400+ bits and remember each stable isotope is indistinguishable from every other atom of that stable isotope.
8 binary bits in order can encode 256 states, as if every bit was capable of encoding 32 states.
Which is why I suggest comparing it to the number of bits required to encode the universe. If all you wanted was to store a single arbitrary number and could read write the universe you could do anything out to 2^(10^(80 * k)) where k is larger than 1 but I suspect below 100.
When people use this phrase they're not trying to claim that the number of atoms in the universe is directly related to the problem at hand; they're just trying to convey scale.
Another argument would perhaps be the energy required to do the computation. Maybe that relates more directly to the number of atoms in the universe, via Einstein's equation?
This is a pretty huge well except. When people talk about information that can be stored in the universe this is exactly the limitation they have in mind.
Measuring distances, I suppose there could be numerous ways, like measuring the gravity or electric pull (not sure what it is called in English). I think only Quantum theory says we can not measure to arbitrary precision, or at least if we do, there are other issue. Still, that would be another argument than simply pointing at the number of atoms.