When will $\text{Variance}=\text{Bias}^2$ hold for the optimal model? Let us consider the bias-variance decomposition in the context of model selection. The picture below suggests the optimal model (the one minimizing the expected squared prediction error) will have $\text{Variance}=\text{Bias}^2$.

This looks like a very special case, as corresponding diagrams in machine learning textbooks are often asymmetric. See e.g. James et al. "Introduction to Statistical Learning" Figure 2.12:

I think the curious result of the initial picture rests on the curves of variance and squared bias being convex and rather symmetric. The convexity is probably sensible, but I am not so sure about the approximate symmetry.
Question: What are some concrete settings (model classes and data generating processes) in which $\text{Variance}=\text{Bias}^2$ could be expected to hold (at least approximately) for the optimal model?
P.S. There is no need to address the question of whether the initial picture must hold in general, as that may lead this thread off track. The answer is clearly negative, as indicated above and in Stephen Kolassa's answer.
 A: 
The picture below suggests the optimal model (the one minimizing the expected squared prediction error) will have $\text{Variance}=\text{Bias}^2$.

Sorry to rain on your parade, but this does not necessarily hold. The picture is misleading. No, that is not an answer to your question, but I would assume this reduces our interest in it altogether...
As an example, assume model complexity $x$ can be parameterized in a single dimension with $0\leq x\leq 1$. Squared bias is given by
$$ b(x) = 1-\sqrt{x}, $$
variance by
$$v(x) = x^2, $$
and total error therefore by
$$ e(x) = b(x)+v(x) = 1-\sqrt{x}+x^2. $$

Bias is decreasing, variance is increasing, both are convex, and the minimal error is achieved at $x=\frac{1}{4^\frac{2}{3}}$, far away from the point of intersection.
The result about minimal error appearing at the intersection should indeed hold if we posit symmetry of bias and variance. But then they would need to be symmetric about some specific $x$ value, and that particular $x$ value would then turn out to be the minimum error complexity. I would not think such a symmetry to be very common, or therefore interesting.
A: 
When will $Variance=Bias^2$ hold for the optimal model?

I think that some ad hoc example can be built but a general rule cannot exist.
First of all is useful to say that the bias-variance tradeoffs (BVT) story matters in prediction only. Read here (What is the relationship between minimizing prediciton error versus parameter estimation error?).
So, we have a true model in one side and several "proposed/estimated" models in another; among the last we looking for the best one.
In general, the level of complexity in the proposed/estimated model that minimize the MSE, depends crucially on the true model. Then, It can seems that BVT story can suggest us that, under a true model with mid level of complexity, something like $Variance = Bias^2$ can hold for the best estimated model (min MSE).
However this perspective do not take into account that the BVT must be analyzed under a precise amount of data. The best proposed/estimated model change if the number of available data change; indeed if we have an infinite amount of data only $Bias$ matters.
Therefore, the relevance of the “symmetry” linked to $Variance = Bias^2$ decrease substantially.
This topic is related
Statistical Learning. Contradictions?
