News sites love studies like this because they can boiled down to a sound bite. Unfortunately people jump to conclusions too quickly. If you actually read the study (I did) it self reports that "future research is needed to understand the causal relations between noncow milk consumption and height." Correlation does not indicate causation. There are several shortcomings with this study. The cohort of non-cow milk drinkers was significantly smaller than the cow milk drinkers. We have no idea why the children not drinking dairy were avoiding it in the first place (were they vegan, were they lactose intolerant, did the parents simply prefer almond milk?) or what the dietary habits of the children were beyond their milk choice. Overall diet could have significantly slanted the results.
It may be possible that avoiding dairy milk simply delays growth, that kids reach normal height after age 3 and catch up to their peers. In this regard delayed growth (and ultimately delayed puberty) could be a benefit of avoiding cows milk as kids today seem to have a much earlier growth curve than they did a few decades ago. There is a lot of "information" online about vegetarian kids being ultimately shorter than meat eating kids (or more specifically milk drinking kids) but most of these are opinion based blogs (like this one) not scientific journals. I can find no long term study supporting plant based diets reducing the height of people over time though I can find a lot of information on how our population has significantly grown in height on average over the past 2 centuries. Vegans are definitely leaner than meat eaters, but this is hardly a concern and may have significant benefits.
The original study hypothesizes that "noncow milk contains less protein and fat than cow milk and may not have the same effect on height." Raising a plant based child carries with it a responsibility to provide a healthy balanced diet. If you choose to raise your child on vegan junk food they will suffer nutritionally. In addition, while it may be appropriate for an adult to attempt to lose weight on a plant based diet, a child should not be subjected the the dieting whims of their parents. To say that plant based milks lack of protein and fat cause short stature is misleading at best. There are plenty of healthy protein and fat sources for vegans and vegetarians. If parents don't understand how to feed their kids they should consult a nutritionist or dietician. Soy milk does contain protein and coconut milk contains fat for those that are looking to avoid dairy, but the stuff you drink in a carton need not be your only source of those macro-nutrients.
When we talk about childhood nutrition as it relates to the body shapes and sizes of our kids it should not be targeted at the exclusion or inclusion of one food item. Rather we need to look at the overall diet of kids, the reduction or elimination of processed foods, sugar, caffeine and fast food. If you choose to feed your child a plant based milk, choose an unsweetened version that they enjoy. Remember soy contains protein so it can be a good choice. The entirety of the diet is a much more important focus ensuring they are eating plenty of fruit, vegetables, nuts, seeds, beans, legumes, etc. Your child needs fat and protein for growth and nervous system health. If they are completely plant based it's essential they get a B12 supplement. As for media headlines, always read them with a grain of salt.
As a side note, the very same journal in the same edition also released a study that "Nonfermented Milk and Other Dairy Products" are associated with all-cause mortality. Let's suppose that almond milk drinkers are shorter (first study) but live longer (second study) based on this one edition of the American Journal of Clinical Nutrition. Which would you choose?