Cross-reactivity is an issue in the ELISA test (the actual antibody test).

But they validated their results with a second step, the "pseudotype neutralisation assay". This test is highly specific for SARS-CoV-2 - as I understand it, they essentially introduce the virus to the hypothetically immune blood samples and then test that it is indeed neutralised by antibodies.

So, I think this Scottish serology test is pretty robust and accounts for these issues with cross-reactivity and specificity of the testing. In addition - they used a control sample from December 2019 none of which tested positive.