Variance as criteria for feature selection
You are correct that variance depends on the scale and typically it is not a good proxy for how informative a feature will be in terms of the response. The exception is zero variance features. A feature with zero variance has the exact same value for all observations and is therefore completely uninformative. Sklearns VarianceThreshold function defaults to removing only the features with exactly zero variance. 

Another group of non-informative features is the near-zero-variance feature. However, typically, classifying a feature as a near-zero-variance feature does not actually rely on computing the feature's variance at all. See, for example, the NearZeroVar function documentation from R's caret package: 


  "nearZeroVar diagnoses predictors that have one unique value (i.e. are zero variance predictors) or predictors that have both of the following characteristics: they have very few unique values relative to the number of samples and the ratio of the frequency of the most common value to the frequency of the second most common value is large."


You can see that the algorithm is not scale-dependent and does not rely on variance.