We present the METADETECTION weak lensing galaxy shape catalogue from the 6-yr Dark Energy Survey (DES Y6) imaging data. This data set is the final release from DES, spanning 4422 deg$^2$ of the southern sky. We describe how the catalogue was constructed, including the two new major processing steps, cell-based image coaddition, and shear measurements with METADETECTION. The DES Y6 M ETADETECTION weak lensing shape catalogue consists of 151 922 791 galaxies detected over $riz$ bands, with an effective number density of $n_{\rm eff}$ = 8.22 galaxies per arcmin$^2$ and shape noise of $\sigma _{\rm e} = 0.29$. We carry out a suite of validation tests on the catalogue, including testing for point spread function (PSF) leakage, testing for the impact of PSF modelling errors, and testing the correlation of the shear measurements with galaxy, PSF, and survey properties. In addition to demonstrating that our catalogue is robust for weak lensing science, we use the DES Y6 image simulation suite to estimate the overall multiplicative shear bias of our shear measurement pipeline. We find no detectable multiplicative bias at the roughly half-per cent level, with $m = (3.4 \pm 6.1) \times 10^{-3}$, at $3\sigma$ uncertainty. This is the first time both cell-based coaddition and M ETADETECTION algorithms are applied to observational data, paving the way to the Stage-IV weak lensing surveys.