Put simply, all diseases are different, that is why they do the modelling. A quick guide to infectious disease epidemiology. The important concept is the base transmission rate, the number of people who a case infects. For COVID-19, it is probably around 3, but may be 4 in a population that is especially sociable or has extended families in the one household like Italy. We will assume 3. Now the important thing is the current transmission rate or what we achieve with interventions. If this is below one, the epidemic dies out. Most countries are currently achieving slightly less than one. So all I need to do with a vaccine is to reduce the transmission rate from 3 to below one. So if 2/3 are vaccinated with a 100% effective vaccine then that should be close enough. The good news is that we already have some strategies to reduce infection that could be continued, and that would make the transmission rate well below one.
Now why the figure of 95%? This is because one of the childhood diseases that is vaccinated against is measles which has a base transmission rate of around 20 (if anyone tells you COVID-19 is highly infectious, it isn’t). So you need around 95% coverage. I assume the vaccine must be close enough to 100% effective or the base transmission rate may be slightly less than 20, because 95% seems to work.