Gaussian Process Regression (fitrgp) regresses rather than interpolates?

I am using fitrgp function to model some noisy data using gaussian process regression. I am also using dacefit function to model the same data using kriging models.

If I calculate the error on the training set, I observe that for kriging, absolute average error on train set is nearly 0 (1e-14) meaning it is "interpolating" on my datapoints, as expected. However, GP regression model shows much larger training error ~1e-2 (if I keep sigma by default and allow it to optimize) which means it is "regressing". Should GP model also not interpolate (like kriging) if I do not assign sigma as 'SigmaConstant', true? Thanks!