The short answer is yes and no. The younger, less experienced the athlete is, the less specific the training has to be. There are some very general skills that need to be learned before any specific work can take place. Given developmental stages and physiological maturity levels, it would be inappropriate to for a program to be sports specific with athletes under a certain age.
To dive into this topic a little more, we first have to look at the difference between a sports skills coach and a strength and conditioning coach (read more here). In terms of strength training, there is some specificity involved with strength exercises and muscle systems that are taxed more than others when participating in certain activities. For example, a soccer player should have an extremely strong squat over an extremely strong bench press; there is just more carry over.
Looking at things from a developmental standpoint, young athletes need to build a big foundation of strength. Athletes need full body strengthening in the global expressions of human movement -- running, jumping, throwing, and lifting. Lifting being a large focus because it will overload the system and allow for more force production which is number one, after sport practice, for improving performance in most sports. At younger ages athletes are weak all around so they need to get stronger…. all around. In my experience, teaching an athlete how to control their body and getting them stronger solves a lot of performance and postural issues. Conditioning, however, can be very sport specific. There are distinct work and rest protocols that can be followed to elicit specific cardio-respiratory and metabolic adaptations, though that is for another article.
In short, a young athlete participating in "sport specific" strength training doesn't make sense in the long run, and most of the time it is just a marketing ploy for a less than ideal program. Unless your athlete is trying to get into college, already in college, or in the pros, "sport specific" strength training is something you should be concerned with. Get strong with the basics first. Strength is a skill; learn it, build it, and play your sport, the middle will take care of itself.
Older PostStrength Coach vs Skills Coach: Two Pieces in the Performance Puzzle.