Well, this has to do with the idea of abduction as a trope in chivalric romance. In these stories, there really wasn’t a big difference between “running off with her” and “kidnapping.” Whether it’s Lancelot and Guinevere or Tristan and Iseult of chivalric romances, or the earlier Celtic sources whether Welsh or Irish (there’s a very similar story that’s part of the legend of Finn Mac Cool), the lady is married and is carried off by the knight against the wishes of her husband (hence why it’s called abduction/kidnapping and not elopment).
Moreover, in a lot of these stories, there are very weird parallels between the very-much-married lady being kidnapped by a mysterious knight and spirited off to a strange castle in the Land of Summer or the Twisted Woods – and if all of this is starting to sound a lot like Faerie, you’re not wrong – from which she is rescued by the gallant knight (which is the spark for their falling in forbidden love) and the lady being rescued by the gallant knight from her husband’s wrath when the adultery is discovered, and spirited off to another castle, which is equally strange.
To bring this around to Rhaegar and Lyanna, the name of the castle that Lancelot takes Guinevere back to is Joyous Gard/Joyeuse Garde. Gard/garde means a keep, a keep is a fortified tower, and thus we get the Tower of Joy. However, Joyous Gard has a deeper significance than just the name: in the beginning of the story, it’s known as Dolorous Gard because the castle has been put under an evil enchantment; Lancelot takes the castle and breaks the spell, but finds within the castle a tomb with his name on it, and knows that he is going to be buried there one day. When Guinevere visits his castle – at a time when she’s still with Arthur and Lancelot and Guinevere are holding to a chaste romance – Lancelot changes the name of the castle to Joyous Gard in her honor. When Lancelot and Guinevere give in to their passion and their adultery is discovered, the castle becomes Dolorous Gard again, suggesting a Fisher King-like situation where the purity of chaste love has become the impurity of carnal love, dooming the land just as their adultery has caused the fall of Camelot, even as the castle becomes the refuge of the lady and her knight.
The point is there’s a lot of doubling: the good knight and the evil knight, the lady fair and the adulterous queen, the good castle and the evil castle, and thus (to get all the way back to your ask) the kidnapper and the rescuer.