Quantum error correction requires accurate and efficient decoding to optimally suppress errors in the encoded information. For concatenated codes, where one code is embedded within another, optimal decoding can be achieved using a message-passing algorithm that sends conditional error probabilities from the lower-level code to a higher-level decoder. In this work, we study the XYZ${}^2$ topological stabilizer code, defined on a honeycomb lattice, and use the fact that it can be viewed as a concatenation of a [[2, 1, 1]] phase-flip parity check code and the surface code with Y ZZY stabilizers, to decode the syndrome information in two steps. We use this sequential decoding scheme to correct errors on data qubits, as well as measurement errors, under various biased error models using both a maximum-likelihood decoder (MLD) and more efficient matching-based decoders. For depolarizing noise we find that the sequential matching decoder gives a threshold of 18.3 %, close to optimal, as a consequence of a favorable, effectively biased, error model on the upper-level YZZY code. For phase-biased noise on data qubits, at a bias $\η =\frac{p_z}{p_x+p_y} = 10, we find that a belief-matching-based] decoder reaches thresholds of 24.1 %, compared to 28.6 % for the MLD. With measurement errors the thresholds are reduced to 3.4 % and 4.3 %, for depolarizing and biased noise respectively, using the belief-matching decoder. This demonstrates that the XYZ$^2$ code has thresholds that are competitive with other codes tailored to biased noise. The results also showcase two essential components in optimizing the performance of concatenated codes: 1) tailoring the upper-level code to the effective noise profile of the decoded lower-level code, and 2) using an upper-level decoder that is adapted to this noise profile.