Are these patients that are admitted with a confirmed 'rona infection, specifically to be treated for the 'rona?
Or, are these patients who were seeking other treatment, and as part of the admission process took a PCR test that generated a "positive" result?
I'll guess the latter
The actual problem with these stats is there is a central federal Repository of information about who has been vaccinated. This was necessary because some people get one shot in one state and another shot in a different state.
Joe Public can’t just pick up the phone and invade the medical privacy of their neighbor. Everyone is not authorized to access that database. This includes Hospital administrators. It can be difficult to verify vaccinations.
The consequence of this is a default. If someone is not confirmed vaccinated they are declared to be unvaccinated. And this can affect the statistic that 87 or 95% of covid patients in a hospital are unvaccinated. It creates unintentional deception. It creates the impression that if you’re vaccinated you will not be in the hospital.
Most states have their own Repository and it is easier to access, but if a patient his visiting family in that state he will not be in that state’s database. This is not a enormous problem because most vaccinations can be verified, but not all and not just 5%. The statistics are likely badly exaggerated for reasons that do not suggest any sort of conspiracy.