Well, I’m going to throw an opinion here, and I’ll base it in part on some of the newer versions, so feel free to disagree.
First, in all of the more recent versions (that don’t suffer nearly as much from filmation’s economizing) Prince Adam and He-Man have distinct differences in physical size and features (size mostly, features are still pretty close), so if we consider those canon, we can leave that part of the argument out.
Second, in the most recent version (Netflix)
Summary
He-Man dies in combat and ends up in the heroic afterlife, where he presents entirely as Prince Adam. And there is no secondary ‘soul’ for He-Man distinctly. However, when he attempts to use the power without the governing Sword of Power, he becomes yet another variant with a more savage appearance, the physical power, but very little of Adam’s compassion. Until when the story demands it, it is awakened by those closest too him.
Massive spoilers, so double blocked.
In short though, I consider He-Man to be Adam, but Adam with a great deal more confidence due to his overwhelming power-up. Cringer is more drastically changed, but there were plenty of episodes where we see quite a bit of latent courage as well, which brings me to my last point.
Over the multiple series, quite a few other people gain the power of Grayskull (though most frequently temporarily) and they also have physical and emotional changes. In many, it’s not quite as dramatic as Adam’s, but it is distinct and includes both emotional and physical changes. Even if it was just the psychological effect of the overwhelming power, it’d be potent, and I’d argue that it works more like a super-amphetamine, where it amplifies existing traits and removes inhibitions (with differing results depending on the underlying personality).
So, rather than considering whether they’re different people before / after a brain injury or the like, a better comparison may be ‘Is a person the same or different after having some awesome shit, like, man?!’
Sure they’re different, but they’re still the same person. Just add a dash of magic, and let the drug actual make you as powerful as you feel when using, and you’re there.
Which was part of the premise in many ways throughout the multiple series, Adam is the best choice to wield the power of Grayskull because it changes his core morality so little.