It's called King & King by Linda de Haan and Stern Nijland. The summary says: When the queen insists that the prince get married and take over as king, the search for a suitable mate does not turn out as expected.
Yes, this basically is every J2 royalty AU that exists, in picture book format. The queen drags a line of eligible princesses through and the prince turns them all down. He's bored and couldn't care less. Finally, one walks in and the prince gets up and says YES! THAT ONE! and everyone is excited until they realize he doesn't actually mean the princess, but instead her brother, who came into the throne room with her. They're surprised, but go with it.
Then the two princes fall in love, get married, and rule the country as King and King. And it truly ends with And everyone lived happily ever after.
I love this book. It's sweet, simple, not preachy in any way, the illustrations are adorable, and it tells a simple story every one, at any age, can grasp.