How is a cadet house born ? Maekar has his own castle ( summerhall) and his son inherits it but he is still a targ. Karlon stark has his own castle but he changes his surname . Thank you

Well, Maekar isn’t a great example because he was the heir of the main branch and then inherited. Also, it’s never been clear whether Prince of Summerhall is a title that one retains for life – I.E, if Aerys I had two sons and the oldest is the Prince of Dragonstone as per tradition, does Maekar cease to be the Prince of Summerhall and the youngest son of Aerys takes the title?

Moreover, the surname doesn’t make a cadet branch either – the Lannisters or Lannisport are a cadet branch, and look at all the Flints all over the North, and so on and so forth. What makes a cadet branch is a separate fiefdom and line of inheritance that persists over time, so that they’re broadly seen as a separate family.