Welcome to UQWorld! I know this has been long posted without a response; I’ll attempt an answer anyway .
I was not previously familiar with the work of Ankenman et al. you mentioned, but I went through it quickly (so I might miss something, and thanks for the reference!). To confirm, there’s no major difference between the formulas presented in the Kriging user manual (Eq.1.16) and the one in the paper (Eq. 6). I think, the formula is a generic formulation for Gaussian process in the case of an additional (or intrinsic, borrowing the term from Ankenman) noise coming from the responses itself. In other words, at the same input point, the output differs every time we observe it (or run the simulation). The textbook by Rasmussen also has the same formulation.
UQLab simply provides support for including such a noise term in the formulation of the Kriging predictor.
The noise term may be the same everywhere (homoscedastic), different but uncorrelated (independent heteroscedastic), or different and correlated (general heteroscedastic). However, it is only in the case of homoscedastic noise that UQLab can estimate the noise variance along with the other Kriging parameters.
Other than that particular case, you need to come up with a way to estimate the noise variance and its structure yourself and then plug it into UQLab.
As you correctly pointed out, the paper of Ankenman provides a procedure on how to estimate the noise variance of a stochastic simulator through replications and sequential experimental design. It seems the procedure is not that intrusive w.r.t to the current implementation of Kriging in UQLab, so it would most probably be feasible to adapt the workflow to implement the procedure (though I cannot say for sure).
I hope this helps!