There are many antibody tests. The one I’ve read about (from the Charité in Berlin) was designed (or found) to also test positive for the “original” SARS, and for very closely related bat viruses (neither of which should present an issue when testing humans today), but not for other corona viruses (such as the common cold etc.).

That test has also very carefully been validated, with excellent sensitivity (=few false negatives) and specificity (=few false positives). Not sure all available tests have gone through quite so thorough validation.