Precise device characterization is a fundamental requirement for a large range of applications using photonic hardware, and constitutes a multi-parameter estimation problem. Estimates based on measurements using single photons or classical light have precision which is limited by shot-noise, while quantum resources can be used to achieve sub-shot-noise precision. However, there are many open questions with regard to the best quantum protocols for multi-parameter estimation, including the ultimate limits to achievable precision, as well as optimal choices for probe states and measurements. In this paper, we develop a formalism based on Fisher information to tackle these questions for set-ups based on linear-optical components and photon-counting measurements. A key ingredient of our analysis is a mapping for equivalent protocols defined for photonic and spin systems, which allows us to draw upon results in the literature for general finite-dimensional systems. Motivated by the protocol in X.-Q. Zhou, et al., Optica 2, 510 (2015), we present new results for quantum-enhanced tomography of unitary processes, including a comparison of Holland-Burnett and NOON probe states.