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Pulsar timing arrays are ensembles of regularly observed millisecond pulsars timed to high precision. Each pulsar in an array could be affected by a suite of noise processes, most of which are astrophysically motivated. Analysing them carefully can be used to understand these physical processes. However, the primary purpose of these experiments is to detect signals that are common to all pulsars, in particular signals associated with a stochastic gravitational wave background. To detect this, it is paramount to appropriately characterise other signals that may otherwise impact array sensitivity or cause a spurious detection. Here we describe the second data release and first detailed noise analysis of the pulsars in the MeerKAT Pulsar Timing Array, comprising high-cadence and high-precision observations of $83$ millisecond pulsars over $4.5$ years. We use this analysis to search for a common signal in the data, finding a process with an amplitude of $\log_{10}\mathrm{A_{CURN}} = -14.25^{+0.21}_{-0.36}$ and spectral index $\gamma_\mathrm{CURN} = 3.60^{+1.31}_{-0.89}$. Fixing the spectral index at the value predicted for a background produced by the inspiral of binary supermassive black holes, we measure the amplitude to be $\log_{10}\mathrm{A_{CURN}} = -14.28^{+0.21}_{-0.21}$ at a significance expressed as a Bayes factor of $\ln(\mathcal{B}) = 4.46$. Under both assumptions, the amplitude that we recover is larger than those reported by other PTA experiments. We use the results of this analysis to forecast our sensitivity to a gravitational wave background possessing the spectral properties of the common signal we have measured.