ZTF SN~Ia DR2: Cosmology-independent constraints on Type Ia supernova standardisation from supernova siblings

Understanding Type Ia supernovae (SNe~Ia) and the empirical standardisation relations that make them excellent distance indicators is vital to improving cosmological constraints. SN~Ia ``siblings", i.e. two or more SNe~Ia in the same host or parent galaxy offer a unique way to infer the standardisation relations and their diversity across the population. We analyse a sample of 25 SN~Ia pairs, observed homogeneously by the Zwicky Transient Factory (ZTF) to infer the SNe~Ia light curve width-luminosity and colour-luminosity parameters $\alpha$ and $\beta$. Using the pairwise constraints from siblings, allowing for a diversity in the standardisation relations, we find $\alpha = 0.218 \pm 0.055$ and $\beta = 3.084 \pm 0.312$, respectively, with a dispersion in $\alpha$ and $\beta$ of $\leq 0.195$ and $\leq 0.923$, respectively, at 95$\%$ C.L. While the median dispersion is large, the values within $\sim 1 \sigma$ are consistent with no dispersion. Hence, fitting for a single global standardisation relation, we find $\alpha = 0.228 \pm 0.029$ and $\beta = 3.160 \pm 0.191$. We find a very small intrinsic scatter of the siblings sample $\sigma_{\rm int} \leq 0.10$ at 95\% C.L. compared to $\sigma_{\rm int} = 0.22 \pm 0.04$ when computing the scatter using the Hubble residuals without comparing them as siblings. Splitting the sample based on host galaxy stellar mass, we find that SNe~Ia in both subsamples have consistent $\alpha$ and $\beta$. The $\beta$ value is consistent with the value for the cosmological sample. However, we find a higher $\alpha$ by $\sim 2.5 - 3.5 \sigma$. The high $\alpha$ is driven by low $x_1$ pairs, potentially suggesting that the slow and fast declining SN~Ia have different slopes of the width-luminosity relation. We can confirm or refute this with increased statistics from near future time-domain surveys. (abridged)