As the game tells us, Emmeryn was attacked by her own people, they threw stones at her, and despised her as well as the entire Exalt family…
For me, that explains why Chrom grew up so strong, yet so dense: he was forced by the situation to grow faster than a normal prince, was raised by random maids instead of loving parents…
He wanted to become stronger to protect his sisters, and help Emmeryn lead the country.
As for Lissa, it justifies how childish she can sometimes be, as Emmeryn and Chrom were older, she had nothing to do in terms of leading or setting the example. The jokes and pranks she planned were to make them laugh or chuckle, to make them forget about their duty, if only for a moment.
When Emmeryn was ten, I like to imagine Frederick was starting his training to become a knight (10yo too). He used to follow a knight close to the royals and protected them as much as he could. Due to being the same age as Emmeryn, they grew up as friends and she often asked for his company when he was available. It was her secret outlet to relieve from her duty.