If the sensitivity/specificity of the test was similar to the paper below, true prevalence for the ship probably approaches 100%.
Probably not. Ships are unusually concentrated social environments.
> These data show that systemic IgA and IgG production against SARS-CoV-2 develops mainly in severe COVID-19, with very high IgA levels seen in patients with severe ARDS, whereas mild disease may be associated with transient serum titers of SARS-CoV-2-specific antibodies but stimulate mucosal SARS-CoV-2-specific IgA secretion. The findings suggest four grades of antibody responses dependent on COVID-19 severity.
But this does not eliminate the possibility raised by the GP, that some people could be clearing even a severe infection with antibodies that are not specific to SARS-CoV-2, and would never get a positive result on a highly-specific test.