The estimation of covariance matrices is a core problem in many modern adaptive signal processing applications. For matrix- and array-valued data, e.g., MIMO communication, EEG/MEG (time versus channel), the covariance matrix of vectorized data may belong to the non-convex set of Kronecker product structure. In addition, the Kronecker factors can also exhibit an additional linear structure. Taking this prior knowledge into account during the estimation process drastically reduces the amount of unknown parameters, and then improves the estimation accuracy. On the other hand, the broad class of complex elliptically symmetric distributions, as well as the related complex angular elliptical distribution, are particularly suited to model heavy-tailed multivariate data. In this context, we first establish novel robust estimators of scatter and shape matrices (both related to a covariance matrix), having a Kronecker product structure with linearly structured Kronecker factors. Then, we conduct a theoretical analysis of their asymptotic performance (i.e., consistency, asymptotic distribution and efficiency), in matched and mismatched scenarios, i.e., when misspecifications between the true and assumed models occur. Finally, numerical results illustrate the theoretical analysis and assess the usefulness of the proposed estimators.