Bad estimates of the number of unvaccinated people can also explain these weird statistics.
Public Health England might have overestimated the unvaccinated population, which means that their estimation of the infection rate for unvaccinated persons is too low.
In addition to potential base rate fallacies pointed out in the other answers above, there might be an even more fundamental measurement error. The estimated infection rate of the unvaccinated population might be off because of bad estimates for the overall population of England.
The More or Less podcast hosted by Time Hartford explains: (manual transcription from the audio source starting around 2:40)
We don't actually know how many unvaccinated people there are in England. Not even close. And that matters a lot because to talk about the infection rate in a group of people, we have to know how many people are in that group. Overestimate the number of people of unvaccinated people and you underestimate their infection rate. And we think that's what Public Health England have done.
To figure out the number of unvaccinated people, we know the number of vaccinated people, which is easy -- we know that from NHS records -- and then subtract that number from the total population. But we don't know the total population of England for sure, and even small changes in the estimate of population lead to big changes in changes in our estimates of the unvaccinated remainder.
More or Less goes on to explain that they believe that Public Health England chose a poor proxy for estimating the total population of England. Instead of using the Office of National Statistics (ONS) estimate, Public Health England used the number of people registered with doctors (NIMS). More or Less argues that this registration number is likely an overestimate of England's population, and that necessarily means that their unvaccinated population is also too large. This leads to the estimated infection rate of the unvaccinated population to be diluted and appear artificially lower than expected.
The reported difference between the two data sources for estimating the number of unvaccinated people age 40-79:
- Based on NIMS data (doctor registrations): 3.5 Million
- Based on ONS population estimate: 1.5 Million
Podcast Guest James Warm (a COVID modeler) explains that using these different estimates for the unvaccinated population changes the report rather dramatically:
... if you use the ONS data [instead of NIMS] to work out that [COVID] case rate, you get a case rate in the unvaccinated which is about double the rate of those who have two doses of vaccine.