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Magnus Carlsen at age 13 was substantially weaker than Kasparov; I suspect the notion they were equals comes from the fact they drew a rapid game, but such upsets like that aren't unheard of (and Kasparov beat him in their next encounter, knocking him out of the tournament).
I'm extremely dubious of this. What's your source?
Starting young (below 10 at a max, ideally below 8) is essential to eventually playing chess at the highest levels. There probably isn't a single player in top 100 who didn't start learning the game before this cut-of (and I suspect 90% of them achieved the GM title before they were 20). The chance to rapidly absorb thousands of patterns while your brain is still plastic is an opportunity you only get as a child. Anyone who's spent much time playing chess will know that children improve dramatically quicker than adults, primarily because they've still got the mental wiring to make great leaps in their understanding of the game in a very short time.
Magnus Carlsen was a prodigy in almost every regard, not just a chess prodigy. His spatial recognition and fluid memory abilities were very advanced at a young age, suggesting a very high full-scale-IQ, not just being specialized at chess.
From wiki:
Kasparov's IQ is widely cited as 135, which is good, but not that impressive. I would bet it's a lot lower than that of Magnus Carlsen.
They faced different competitors. Given how optimized chess has become, Magnus Carlsen faced harder opponents and a much deeper talent pool. In 2004, when he lost to Kasparov, Magnus was just 13. It would be another decade until he would become the world chess campion and hit the peak of his abilities. A more apt comparison would be adult Magnus vs Kasparov.
Carlsen has also benefited from the advances in theory and training methods that have occurred over the last few decades. If Kasparov were playing today and were able to take advantage of these resources he'd most likely be even better than he was at his peak.
That's pretty open debate among chess players with plenty of people taking either side. Neither is generally accepted by a majority of players to be more talented than the other.
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