<p>This paper investigates the identifiability of structured systems with a matrix fraction description and attacks practical difficulties in estimating parameters under finite frequency responses. The numerator and denominator polynomial matrices depend affinely on system parameters, which can represent a broad range of systems, especially circuits and mechanical systems. Except for assuming that the denominator matrix is invertible, no further assumptions are adopted. A criterion is provided for validating this assumption utilizing graph theories and matroids. A sufficient and necessary criterion for global identifiability based on the rank of a numerical matrix is given and can be recursively verified with each frequency point, which is computationally attractive for large-scale systems. For an identifiable system, an ellipsoid approximation is obtained for the set of parameters whose corresponding system frequency responses deviate within a specific range from those of the system at a specific parameter value. Using this approximation, explicit expressions for absolute and relative sloppiness have been derived, which can be used to quantify the difficulty of identifying system parameters. Comparisons with the well-known Fisher information matrix have also been performed through a mechanical system, revealing significant differences in quantifying parameter estimation hardness.</p>