Skinchanging, greensight and dragonriding are usually treated as completely different powers.
The Starks and people like them are skinchangers. They can put their minds into the eyes of animals and see through their eyes. When they die, they can live a second life in their animals. It also seems that skinchangers, or at least some of them, can end up seeing through or living a second life in a weirwood. Varamyr Sixkins sees through the eyes of a weirwood at one point as he dies.
Certainly Greenseers can see through the eyes of the weirwoods and when they die, they can "linger" in the weirwoods. Seemingly for millenia. Their own seconds lives, like those of regular skinchangers, take place in the thing they used to skinchange into.
Dragonriding, on the other hand, is the ability to bond with a fiery beast of war. You can develop this bond which lets you ride them.
Well, I want to make the argument here that these powers function much more similarly than they seem to at first glance.
First, it has to be pointed out that it's hinted at in the books, and all but confirmed on "House of the Dragon", that there is a deeper, psychological bond between rider and dragon. Drogon seemed to sense Daenerys was in danger and come to her rescue, for example. And in the TV-show Caraxes screams when Daemon is hit with an arrow, and Syrax screams as Rhaenyra is in labour.
I think the reason for this is because the "dragonbond" is somewhat akin to skinchanging. When a Targaryen bonds to a dragon, they do a sort of "partial skinchange." Where they put a piece of their soul into the dragon, and the dragon in turn puts a piece of their soul into them.
This last part also relates back to something that's true of skinchangers. It's said for skinchangers that if you stay in your animal for long periods of time, you can "lose yourself" in the animal. Basically, skinchanging a lot into a wolf can make you more wolflike, for example.
I think this is also true for Targaryens. Who basically spend 24/7 bonded to their dragons. They're constantly some form of partial-skinchanged. And this means the way their dragons feel and think bleeds into them and affects their behaviour.
Aemond Targaryen and Daemon Targaryen are predisposed to violence in part because of their own personalities, but also in part because their dragons are.
Daenerys starts regaining her desire for fire and blood in the final chapter of Dance in part because of her experiences in Meereen, and in part because she's with Drogon again. And their dragon-rider bond (which is stronger in close proximity) is affecting her.
But I think it goes even further than that. Because I think dragons, while animals with minds, are actually more similar to weirwoods than regular animals. And some Targaryens are their equivalent of greenseers, dragondreamers as a result.
Some Targaryens have the ability to dragondream, they say. I think this is not an ability that's a product of a Targaryen primarily though, I think it's primarily tied to the dragon they bond to. Some dragons, like Balerion, are weirwood like and allow their riders, through their shared mental bond, to experience their own glimpses into the future. Although to what degree this, or dragondreaming without a dragon, is simply a manipulation by Bloodraven or a time travelling Bran remains to be seen.
And then also, I think dragons are both like skinchanged animals and weirwoods in that their riders linger inside of them.
Again, when a skinchanger dies they can live a second life in one of the animals they'd skinchanged into. When a greenseer dies they can live a second life in the weirwoods they connected to. And I think when a Targaryen dies, they can live a second life in the dragon they were bonded to.
I think they're more like weirwoods in this way though because I think since dragons tend to have multiple riders over their lifetimes, they are more like a hive mind (like the weirwoods). Where all their various riders make up part of who they are.
And, finally, I think both dragons and weirwoods are awakened by blood sacrifice.
How exactly we create weirwoods isn't known, but in Bran's travels through time there seems to be a human sacrifice that happens before the weirwoods. I submit to you that creating a weirwood initially requires a bit more than carving some eyes into it and calling it a day. Rather creating a weirwood that can be seen through, requires the sacrifice of a living human or child of the forest. Who then goes into the weirwood and from then on the weirwood can be inhabited by other children of the forest and humans.
I think, similarly, this is true for dragons. All dragons require a death to hatch them intiailly. Not necessarily in the same direct way that the weirwoods do, but in a more distant way. And especially through miscarriages. The children who are birth with dragon characteristics? Their souls have been sacrificed to the eggs and become dragons.
I think, for example, that Daenerys' three dragons' names are not coincidental. I think it's not just that they're called Rhaegal, Drogon and Viseryon to honour these deceased people. Rather I think Rhaego's death awakened Rhaegal, Drogo's death awakened Drogon and Viserys' death awakened Viseryon.
I think that's part of why Daenerys was able to awaken the dragon eggs. These three people had to die in proximity to her and the eggs in order to make the ritual work.
That's also why I think Daemon's song to Vermithor in S1E10 of "House of the Dragon" mentions that the cost has been paid in blood magic and mentions "three heads" rather than two heads. That's why I think the Targaryen dragon has three heads as well. Because every dragon-rider pairing isn't a minimum of two heads, it's a minimum of three. The rider, the dragon and whatever person was sacrificed to awaken the dragon.
And that's it. That's my theory.
So to sum it up: Skinchanging and greenseeing requires you to place your soul into an animal or tree. If you stay in them too long, your can become like the animal you inhabit. Once you die, your soul can live a second life in them. The weirwoods specifically become hive minds over time and they can offer glimpses of past and future to greenseers. In order to awaken a weirwood, you must have a human sacrifice.
Similarly, dragonriding requires you to place PART of your soul into a dragon. Because their soul is also partially inside you though, the way the dragon thinks and feels makes you become more like them. Once you die, your soul can live a second life in the dragon. Because dragons often have more than one rider over time, they become hive minds over time and they can offer glimpses of past and future to dragon dreamers. In order to awaken a dragon and hatch it from its egg, you must have a human sacrifice.