The blood test looks for antibodies, so it detects whether the person has had an immune response to the virus.

The GP is pointing out that having an immune response to a first infection has not been riguorsly demonstrated to prevent future infection. If we are unlucky, immunity may only persist for a short period of time.

I believe in those "recovered patient tested positive again" cases, they are talking about PCR testing (which detects virus RNA), not blood antibody tests.

Okay, I see what you are saying.

Both things can be true though, we don't know how persistent the immune response will be.

Antibodies (Immunoglobulins) will fade over time, and may not be detectable in the blood after a while. So a negative antibody test for Covid-19 wouldn't necessarily mean you're not immune.

However, immunity should persist for a long time, perhaps even for life. That's because the immune system retains memory of past pathogens and knows how to make new antibodies quickly if needed.

The reason we seem to keep getting infected with cold and flu, year after year, is because these are actually different viruses each time.

You are carefully explaining the mostly true generality.

The fact remains, we simply don't know how long immunity will persist for a given novel virus. We will find out over the next months and years, but at this time, we do not have concrete knowledge about SARS-CoV-2.

Which blood test currently in use looks for antibodies?

It's explained on the page I linked earlier in the thread. A founder of the company that developed it is paying to have the county where he lives tested.