I'm a fan of mundane reasons, and of calling elves and dwarves "fantasy species" or "fantasy hominids" instead of "races." Here are several mundane reasons for similar growth and maturation rates across your fantasy hominids:
Historic Hominid Growth Rates
Very clever researchers have examined the growth rates of ancient hominids. They've found that there are growth rates for hominids are all over the place! Some of them grew just as fast as modern humans, whereas others grew faster or slower. Another paper claims no correlation between body size and growth rates. The growth rates of certain variables do seem to be shared between certain species, but nothing so simple as "a creature can expect 30% of their life to be spent as a youngster."
It seems that Bonobos, Orangutans, and Chimpanzees are giving birth for the first time at around 14 years old, at have a maximum age around 55. (Compared to the human hunter-gatherers, who can give birth at around 19, and live to be 85.) Given the number of related species who have their first children around the same time, the second paper mentioned concludes that the ancestral hominid likely reached sexual maturity around age 13 to 15. If your fantasy world assumes some sort of evolutionary connection between humans and other fantasy races, this baseline of sexual maturity around 14 should be kept. Obviously, this extended pubescent period found in humans could be carried over to these fantasy hominids.
Factors of Growth Rates
There do seem to be some common factors which help determine growth rates:
- Food: the type of food and how much was available seems to determine how quickly a species can grow. The more calories you get from your food, the more you can put into growing, surviving, and rearing offspring.
- Total Body Mass: The more massive a creature is, it seems it takes longer to get that massive, given a similar diet compared to another creature. It also indicates that the babies of more massive specimen are more healthy, simply because the larger animal can expend more resources on their child.
So why would a species want to mature quickly, and then go ahead and have a long time spent as an adult? Kin Assistance is an answer here. The idea is that longer lived, possibly infertile, adults can provide assistance to those younger than them. This assistance can guarantee their offsprings' survival, making it a good evolutionary strategy.
Alternatively, older individuals can have some better advantage in the make-babies race of evolution, so it benefits to be a physiologically older individual than not. This means that those who grow up fast can get to make more babies, resulting in babies which grow up fast, so they can make more babies. You see this in lobsters, but this idea is part of the core of the Patriarch Hypothesis, so you can see this in hominids.
Another answer can be found in that babies are are a resources drain. If two species are in competition, and the main difference between them is maturation rate, the species who matures more quickly can defend and out compete the other. The slower-maturing species would be spending resources on their babies while the other is doing other things, possibly with more capable adults. This is not quite the same as the Patriarch or Grandmother Hypothesis, but it is a good reason why dwarves and elves should grow up about as fast as humans. This is especially true when fantasy hominids compete for resources they all need.