As far as you use the cosine as similarity measure, the matrix is a correlation matrix. For this situation in statistics there is the concept of "canonical correlation", and this might be then the most appropriate for your case: it gives an index how much "variance of one set of variables is explained by the other". The two set of variables are the two sets of vectors $\small v_i $ here.

Another option could be to compute the cholesky loadings L1 and L2 of each of the correlation matrices R1 and R2 and do a target-rotation of L1 to L2. Then, for instance, the squared distances of the vector-tips of each related vector in rotated(L1) and L2 could be summed and this could be understand as similarity measure of the matrices(!) - but this is no standard method as far as I know...

As far as you use the cosine as similarity measure, the matrix is a correlation matrix. For this situation in statistics there is the concept of "canonical correlation", and this might be then the most appropriate for your case: it gives an index how much "variance of one set of variables is explained by the other". The two set of variables are the two sets of vectors $\small v_i $ here.

Another option could be to compute the cholesky factors ("factor loadings matrices") L1 and L2 of each of the correlation matrices R1 and R2 and do a target-rotation of L1 to L2. Then, for instance, the squared distances of the vector-tips of each related vector in rotated(L1) and L2 could be summed and this could be understood as similarity measure of the matrices(!) - but this is no standard method as far as I know...