Some readers find it upsetting that the princess of The Princess and the Pea is valued according to shallow standards of wealth, which in the fairy tale crosses over to a ridiculous test of "true" princesshood. Maria Tatar suggests that "the sensitivity of the princess can also be read on a metaphorical level as a measure of the depth of her feeling and compassion."
When I first read that, my reaction was, "eh, that's a stretch," but on further reading of the tale I totally agree. I must admit this is one of the tales I haven't read in a while (if ever...? It's not in my collection of Andersen tales for some reason) and assume I know enough just from being familiar with the plot. Many people tend to make assumptions based on familiar plot points without actually reading the full stories, which can often surprise you...
1. "He went in search of a princess of his own, but he wanted her to be a true princess. And so, he traveled all over the world in order to find one, but something was always wrong."
If by "true princess" the prince just meant a girl born to a king and a queen, it should not have been that hard for him to find one in his travels around the world. This stipulation sounds to me more like when adults refer to little girls as princesses, either to increase their self esteem or motivate them towards more noble behavior (like George MacDonald in his introduction to Princess and the Goblin).
2. One evening there was a terrible storm, and "there was a princess standing outside. But goodness gracious! What a sight she was out in the rain in this kind of weather! Water was running down her hair and her clothes. It flowed out through the tips of her shoes and back out again through the heels. And still, she insisted that she was a real princess."
The famous rest of the fairy tale, with the pea and twenty mattresses, is probably meant to be humorous and whimsical. The pea itself goes on to fame, on display a museum, "where it is still on display, unless someone has stolen it.
There. That's something of a story, isn't it?"
Illustrations by Kay Nielsen