The fact that sixteen years into Spike's life and everyone still calls him a baby dragon is what shows that dragons age differently. It's not implied, it's strictly stated by the show and all related material. Not to mention that in general dragons are shown to age differently in all other fantasy worlds: Lord of the Rings, Game of Thrones, Elder Scrolls, you name it. Dragons can usually live for hundreds to thousands of years, thus each stage of their life (baby, toddler, kid, teenager, adult) lasts far longer than other species.
You're comparing Earth to Equestria. Spike is a baby dragon not a baby pony, thus he is given more duties. He might be a baby, but he is a sixteen year old baby that can do more than a baby pony can.