Based on this quote: "Creates models and generates predictions using an adaptation of Leo Breiman's random forest algorithm" from the ESRI function documentation I would not assume that the algorythm is equivalent to the the original published method.
Unfortunately, the documentation does not state what type of "adaptation" is implemented but, in my experience ESRI does all kinds of hinky things. At first blush it seems like there are all sorts of things going on behind the scenes, as far as how the spatial data is digested into the model. In addition, it looks like they are using Strobl's Conditional variable importance parameter selection method which, I have found is not appropriate in ecological models or remote sensing classification.
What is not clear is if they have also implemented Strobl's proposed alternative node splitting approach. Where Strobl's work is completely valid, it was focused on genetic array data and their simulations were structured around nominal (binomial) responses and not continuous data. As such, their proposed node splitting algorythm and conditional variable importance measures are predicated on a combinatorics data structure of nominal indicator (x) variables. One must also keep in mind that, from any inferential or result(s) based model evaluation criteria for a dependent variable (y), their end result was the parameter selection itself, indicating gene expression, and not any type of prediction of a response variable. In relation to the application of Random Forests models in other disciplines (particularly ecology and remote sensing), this research is often taken way out of context and applied in very unintended ways.