There is always more to be said, but you may be interested in /u/Welfontheshelf's response to How did medieval European monarchs handle twins?

Here is an answer I made (and edited) to a similar question posed sometime ago:

The issue of twins as heirs to throne rarely arise in monarchies, and when they do, the families often resort to the rules of primogeniture to decide who inherits the throne. The rules differ widely, depending on the family, the country and the point in history.

That being said, although cases of firstborn (in the sense that they are the first children of the monarch) twins are rare and there are only a few instances of it occurring throughout history, they are certainly not non-existent.

A good example would be the case of the former Shah of Iran, Mohammed Reza Pahlavi who had a twin sister, Princess Ashraf Pahlavi. In this case, there was really no question of Ashraf ascending the throne, since she was born five hours after her twin brother. More importantly, women were not allowed to ascend the throne of Iran due to agnatic primogeniture which excluded females and female-line descendants, so it fell to her brother to be the heir.

A more interesting (albeit recent) example is the case of the current Hereditary Prince of Monaco, who was born minutes after his sister. However, since Monaco practiced male-preference primogeniture, meaning that males are preferred to inherit the throne than females, it was the younger male twin who was declared the heir and not the older sister.

To sum up, the answer to this question greatly varies on the type of monarchy to which the twins are born. In current monarchies like the Netherlands and Norway, which practice absolute primogeniture (which means that the eldest offspring, regardless of sex will inherit) then it would be the first child out of the womb. However, if a monarchy practices male preference primogeniture (meaning that the eldest son will inherit, regardless of the existence of elder sisters) or adheres to Salic Law (meaning that females are absolutely barred from inheriting the throne, as in the case of the defunct French royalty) then the sex of the twins/triplets will be controlling. If all of them are the same sex, then the eldest will be proclaimed the heir. There were even instances of monarchs being able to choose their successors, disregarding the traditional line of inheritance. Such an example would the Russian Monarchy before the promulgation of the Pauline Laws which clearly defined the position of heir to the throne. Before the imposition of such laws, the monarch could choose his successor and theoretically, could choose either (or both?) of the twins to inherit. Again, this hypothetical.

The reason we have to resort to conjecture and hypothetical situations is that situations of twins or triplets rarely arise in royal families, and less so with them being the first born offsprings of the monarch. The few examples I have mentioned above are from recent history.