How are antibodies specific for a disease detected in the blood if everybody produces a different antibody for the same antigen?
Specific antibodies are typically detected using ELISA.

The way you make a test for an antibody to a particular pathogen is not by using secondary antibodies to the specific part of the target antibody, but by using the antigen.

Different kits work differently, but the general idea is that you have antigen stuck to a surface, you put the sample on that surface. If there are antibodies to the antigen, they stick to it. Then you wash everything away and add a tagged antibody that binds to generic human antibodies.

See also https://www.sciencemag.org/news/2020/03/new-blood-tests-antibodies-could-show-true-scale-coronavirus-pandemic