I recall this episode. This school was some sort of Battle Academy, and it's students were obsessed, and had tunnel vision, regarding the statistics of Pokemon. They would train and capture Pokemon based solely on competitive value, not on personality or love as Ash and the majority of the characters tend to do. They even had their own battle simulator they predict the outcome of battles. This is why they would speak frequently of a Pokemon's value, attacks and power, as they would see Pokemon as components in a battle. However, as most anime characters treat their Pokemon as dear friends,it is rare in the anime that level and statistics play an important role. Of course, when Ash and the gang proved to them that Pokemon can be both great companions and great battlers by beating their leader in a battle, they changed their ways.
Basically, the students mention levels and attacks frequently because their school was focused on the competitive training of Pokemon, while most trainers have Pokemon as companions on a journey, which is why attacks and level are rarely mentioned otherwise.
Hope this helps.