We don't see any other Spikes running around anywhere in pony society, so as I see it we have three possible scenarios:

1. The "hatch egg to get into school" thing only happened once, other candidates were tasked to do other stuff

2. Before Twilight, no candidate was ever able to hatch the egg. It probably wasn't even necessary to actually hatch the egg, it was only important to see how the candidate tries (see Kobayashi Maru)

3. Dragons hatched as an entrance exam don't usually stay with the one who hatched them. They might be teleported back to dragon lands as babies before they can remember anything for example.