A better term for $R^TR$ would be dot-product matrix, which is what this book chapter calls it, cf. equation (6). The matrix entries of the product $R^T R$ are exactly dot products between columns of $R$. It would be pretty hard to make a connection with the cross-product of vectors in this construction.
The usefulness of $R^T R$ lies in its being symmetric and positive semi-definite for any real matrix $R$, not necessarily square. It can well be called a covariance matrix because of its application in statistics, but the construction also appears in least squares solutions of overdetermined linear systems.
Working with the (real) symmetric matrix $R^T R$ allows us the luxury of having a full basis of eigenvectors, and the corresponding eigenvalues will be nonnegative since $R^T R$ is positive semi-definite as well (indeed, positive definite if $R$ has rank equal to its number of columns).
This amounts to a way of defining the singular values of $R$ as (principal) square roots of the eigenvalues of $R^T R$. Another term for essential the same technique is principal components analysis or PCA, which may be more commonly used in statistics and signal processing.