No, The Register has misrepresented the story.
There are several parts to this question. The summary is:
"Peak Wind" is a myth: there is nothing similar in wind to how hydrocarbon stocks will get to such a depletion point that annual production will decline inexorably thereafter.
The modelling methodology smackdown - Have previous estimates over-estimated the global onshore wind resource? This is currently unanswerable. There are two conflicting modelling methods - the aggregated micro-scale; and the meso-scale. The meso-scale modelling in this new Adams & Keith paper (and in the other related papers by Lee & Keith, and the Lee & Kleidon paper in Davephd's answer) cannot yet be validated, because there is insufficient information about the performance of wind farms greater than 100km2 in area - we haven't built them yet. Until we have real-world data on very large wind farms, we cannot say which modelling method is more accurate.
Does it matter? No, it doesn't. There is a wide range of different estimates of global onshore wind resource, that are derived using different modelling methods. They all agree that the global onshore wind resource way exceeds global electricity demand, many times over. And in the absence of validation of meso-scale modelling, it will not get used for wind-farm design, which will continue to use micro-scale modelling.
"Peak Wind" is a myth
The original journal article is here. The concept of Peak Oil, which Peak Wind is being held analagous to, is the simple notion that when depleting a finite non-renewed resource, there comes a time when the extraction rate reaches a maximum, and thereafter declines year on year.
The paper by Adams & Keith that The Register claims to be reporting on, makes no such claim for wind: there is no claim in the paper that there is a wind resource which will become exhausted. That's because wind is constantly renewed by incoming solar radiation.
A typical Peak Oil peaks then depletes. Whereas global wind generation continues to increase as global capacity increases.
Asymptotic wind production
The paper in question models the output of power per unit area, and predicts that it will level out at around 1.2W/m2. Which means that with increasing wind capacity installed, wind output would continue to be a rising curve, and would not show a peak and a down-turn.
Instead, the paper is claiming that there are diminishing returns: that is, that building more wind will continue to yield more electricity, but at a declining rate. This is something that has been known for a long time. The difference is that Adams & Keith are claiming that it will happen faster than previously modelled, and that the global potential onshore wind resource is only a few times global electricity demand, rather than many times it.
Their specific testable claim is that:
wind power production is limited to about 1 W/m2 at wind farm scales larger than about 100 km2
As yet, we have very few windfarms larger than that, so this remain a theoretical exercise. There are plans to build windfarms substantially bigger than that over the next 10 years, so we will find out soon enough.
The modelling methodology smackdown
This paper is part of a long on-going discussion in the literature about the most appropriate way to model the wake effects of large onshore wind farms. The difference boils down to whether one models from the wake of an individual turbine upwards (the Jacobson method of aggregating from the micro-scale), or whether one uses
a parameterization of the atmospheric effects of wind turbine arrays
as Adams and Keith do in the paper in question (and as Keith & Lee do, and Lee and Kleidon do). This is to a degree a question of faith, as to whether the parameterization, in throwing away the detail of the individual turbine, adds to, or detracts from, the usefulness of the model. It's a question of faith, because as yet we don't have enough data from wind farms over 100 square kilometres to validate the meso models.
It's an interesting discussion about the fundamentals of atmospheric modelling, not least because Mark Z Jacobson is the man who literally wrote the book on it. His papers on the subject are listed here.
Does it matter for (inter-)national policy?
There are no implications for (inter-)national policy. Earlier estimates are of a global wind resource at least an order of magnitude greater than global electricity demand, so a quartering of that resource (as Adams & Keith predict) still results in potential global onshore wind production exceeding global electricity demand. That is to say. there is still no meaningful limit on the onshore wind resource. And that's ignoring the massive potential in the offshore wind resource.
Does it matter for wind-farm design?
As of now, no it does not: individual wind farms still get designed using wake-modelling at the level of individual turbines, rather than relying only on Adams & Keith meso-scale modelling. If there comes a time where empirical evidence supports their parameterised meso-scale modelling, then that may get used, because it's computationally simpler, faster and requires fewer inputs. Onshore wind farms larger than 100km2 will get built in the coming decade, so the testable hypothesis I mentioned above will get tested.