Argument mining has become a popular research area in NLP. It typically includes the identification of argumentative components, e.g. claims, as the central component of an argument. We perform a qualitative analysis across six different datasets and show that these appear to conceptualize claims quite differently. To learn about the consequences of such different conceptualizations of claim for practical applications, we carried out extensive experiments using state-of-the-art feature-rich and deep learning systems, to identify claims in a cross-domain fashion. While the divergent perception of claims in different datasets is indeed harmful to cross-domain classification, we show that there are shared properties on the lexical level as well as system configurations that can help to overcome these gaps.