Apologies if I am not too clear, you may use England and Norway as an example of what might have happened in such a scenario.
That's a very common situation. For instance Queen Elizabeth is monarch of both the UK (not England, it's specifically the UK) and Canada. These are two separate crowns, and Canada could chooses to become a republic (boring) or to alter the rules of succession so that they do not align with the U.K.
Nothing weird happens it's just a personal union.
If it's deemed undesirable one or both kingdoms would alter sucession laws.