And the vaccine is about 95% effective, which means around 40 people not immunized. This is meaningless.

No. It is not meaningless. Had the vaccine had the same efficacy on the various variants, the ratio between them would have been the same. However it is not, indicating that the vaccine is vastly (factor of 8) more effective on the main variant compared to the SA / Brazil variants. Hence the conclusion.

It could do with more data but that has also been the case for similar studies of the AstraZenaca vaccine.

Wouldn’t you need more than 800 people to get a statistically significant result?

They check for significance in the paper. There is no magical sample size number. It depends on the size of the effect which here appears to be quite strong so a smaller sample would actually do.

There's a factor of 8 because there are 8 variant cases. This is meaningless.

Is vaccine effectiveness per person or per infection opportunity?

In other words, does 95% effective mean that if you took a large group vaccinated people and once a day had an infected person have a massive coughing and sneezing fit right into their faces, 95% would never get infected by that and 5% would?

Or does it mean that they would all get infected, but it would take on average 20 times as long as it would have had they not been vaccinated?

It means that in the study submitted to the FDA, the vaccine group had 8 cases and the control group had 162 cases [0]. 8 / 162 = ~5% of the cases in the vaccine group, ergo 95% effectiveness (or more specifically Relative Risk Reduction).

[0] https://www.nejm.org/doi/full/10.1056/NEJMoa2034577