Think that in tipical medieval, feudal times, the King was just the primum inter pares (first among equals). He would not have direct control but over his own personal fiefdoms, for the rest of the country he would rely on the loyalty of other nobles that were his vassals (but, if those were powerful enough, could decide to ignore or even oppose him).
In this situation, usually the King would be unable to change the institutions, and the mobility would have wanted to keep their power.
So, the concept of "merging" kingdoms (or duchies, principalities, whatever) would be alien to those people. What you would have would be the same king holding simultaneously the titles to several entities, but each of them would remain separated. Spain it is a good example, after Isabel and Fernando the kings were always common, but the kingdoms of Castille and Aragon did exist, did have different institutions and laws, and each new king would need to be confirmed as king of each kingdom. And we are talking here about the Modern Age (with formal unification coming only after the Spanish War of Succession, 1714), when the power of kings was considerably increased in relation to the nobility.
As a result, kings and other nobles would amass long lists of titles covering all of their rights to the land they held (and some that they did not held but that they did claim), and that they had got through inheritance or conquest.
That does not mean that the nobility of a country would be uninterested in their king becoming king of other country, but it was... complicated. On one hand, it could mean that these nobles maybe could use the king's influence to get holdings in the new country. In the other hand, it could mean that the king could try to drag them into supporting his campaigns to defend his new kingdom. Or that the king could develop a new support base in the other country and use it to be less dependent on local nobility.
As a side note, perhaps the Polish-Lithuanian Commonwealth is a better study case, since as stated elsewhere, the "Spanish" personal union was too late to be considered medieval.