Jon Snow from Game of Thrones is a Targaryen. Does that mean he's the actual ruler of Dragonstone now? Could Jon have been made ruler of Dragonstone due to his heritage?
But not as a Targaryen.
As a rewards for his services.
It is customary at the end of a war to do the Title Dance.
Enemies are stripped from title, land and castles (if let alive at all), while allies and important persons (Sir Bronn of the Blackwater for example) are given titles, land and castle that are currently available, either because the family died in war (Tarly house for example, in fact the all reach is mentioned in the show to be available, and of course, Harrenhal the hot potato), or because it was stripped from enemies.
So he can get any Castle he wants as long as he asks nicely.
Could Jon have been made ruler of Dragonstone?
- Before S08E06, he could have asked Dany for Dragonstone, and she probably would have granted his request.
Isn't he the actual rule of Dragonstone now (in exile)?
If Dragonstone is passed to the heir by birth, then yes - Jon is the last remaining Targaryen, so he would inherit Dragonstone.
But - as you say, he is in exile. So as a member of the Night's Watch, he is unable to inherit property at all.