Adoption would be available to all MCs. While the adopted child would certainly consider both monarchs as parents, they would have only one surname and end up ruling under only one dynasty. This is strictly speaking outside the scope of the game, since the game takes place over several weeks only. However, it would still be influenced by the degree of cooperation and trust your MC fosters with their consort. E.g. if you marry the true Kidia heir and agree to a joint rule (instead of pushing them into the role of consort), an adopted heir would most likely assume the Kidia name and continue the Kidian dynasty.
It’s things like these that the compromising/autocratic trait was created for.
And of course, if you hypothetically end up having several children of different provenance, that’s a recipe for an inheritance dispute. Which again is outside the scope of this game, but worth thinking about nonetheless.