3D-printed medical prototypes, which use synthetic metamaterials to mimic biological tissue, are becoming increasingly important in urgent surgical applications. However, the mimicking of tissue mechanical properties via 3D-printed metamaterial can be difficult and time-consuming, due to the functional nature of both inputs(metamaterial structure) and outputs (its corresponding mechanical response curve). To deal with this, we propose a novel function-on-function kriging model for efficient emulation and tissue-mimicking optimization. For functional inputs, a key novelty of our model is the spectral-distance (SpeD) correlation function, which captures important spectral differences between two functional inputs. Dependencies for functional outputs are then modeled via a co-kriging framework. We further adopt sparse priors on both the input spectra and the output co-kriging covariance matrix, which allows the emulator to learn and incorporate important physics (e.g., dominant input frequencies, output curve properties). Finally, we demonstrate the effectiveness of the proposed SpeD emulator in a real-world study on mimicking human aortic tissue, with improved performance observed.