“Can Discord even do that?  I know he can make ponies with wings or horns into earth ponies, but a swap to changeling? Is that possible?”

Discord has been shown in canon to alter the species of the mane 6 with a snap of his fingers and a whim.  He’s been shown to alter reality, access and interact with different dimensions.  There is no defined limit to his capabilities in the show’s canon, so long as the actions or consequences cause some extent of chaos.

If Twilight can turn 6 grown ponies into Breezies with relatively little effort, Discord can turn one filly into a changeling princess so that she matches the species of her natural mother, especially if it causes some measure of chaos.