The thing about antibodies is that they usually have some degree of cross-reactivity. That means that they recognize the target and also other things that are similar to the target. An analogy would be reading lowercase L as the number 1. So, in this case, I suspect that the antibodies are reacting to coronavirus, not specifically Covid-19. Coronaviruses are fairly common.
President Trump said the other day that an antibody test is going through testing for FDA approval. That means that it will have to meet certain standards of specificity and sensitivity. Even then, I’d still be careful about interpretation of test results.
I think you are suggesting that the two people I mentioned who tested positive for antibodies have gotten a false positive. That may be. One of them works in a medical testing laboratory. Whether that makes your suggestion more or less likely is not something I can assess.