Yes, this was very much a thing.

I can give some famous examples. But, before that, I think it's important to realize what medieval warfare was all about. Battles were rare. Battles were expensive and you could never guarantee the outcome especially at a time where technology was about equal. When there was a battle the nobility were rarely killed by the other side. It was very much the custom to take nobility hostage and ransom them off. Battles were costly things and the nobility fighting would want to make a profit off of it. The habit of taking royalty hostage was just an extension of this custom.

By contrast, the bulk of the non-noble fighting men could be slaughtered.

Henry V famously had all the prisoners slaughtered at Agincourt, which prompted outrage from his own people. Not because of the cost of human life or their love of their fellow man. Because of all that lost ransom.

Hostages were usually taken after battle but not exclusively. Hostages were taken to ensure "good behavior" from the other side, with varying levels of effectiveness. A common practice through the modern era. Important personages were kidnapped when opportunity arose for profit or to get advantage in a war.

So, famous examples:

Perhaps the most famous example of all time is Richard I the Lionheart. King of England and a Great Crusader. He was taken captive on his way back from crusade in the Holy Land. Initially he was captured by Duke Leopold of Austria (whom Richard had insulted while on Crusade.) He was then transferred to the captivity of Leopold's overlord the Holy Roman Emperor, Henry VI. This was an outrage on a lot of fronts. Those who had taken the cross on crusade were supposed to be protected. The Pope was outraged and excommunicated Leopold and Henry VI. But Richard was not released until Richard's mother had amassed an enormous sum (150,000 marks) for ransom.

Not only did this not lead to war but Richard I and Henry VI were later allies against France.

There was also Charles II of Naples who was captured in battle by the forces of Peter III of Aragon. Peter III was actually of a mind to execute Charles II but his wife intervened. Charles was not King when he was captured but became king during his captivity due to the death of his father (Charles I/Charles of Anjou.) He was ransomed for 50,000 marks after a long negotiation lead by Edward I of England.

The English under Edward III were able to capture the French king John II during the Battle of Poitiers. Under a treaty John II was permitted to return home to France to raise his own ransom and his son Louis was held in his place. Louis managed to escape captivity. John, horrified by the dishonor of breaking his oath, willingly returned to England as prisoner. (The exact why he returned is a matter of speculation. Chivalry was idealized to an extreme at this point but no one really would have held it against him had he stayed in France.) Regardless, he ended up dying in captivity.

Going into the Renaissance era, Francis I of France was held captive by Charles V, the Holy Roman Emperor. To ensure his release, Francis I had to make land concessions. And turn over his two young sons (princes) to be held hostage in his stead. As soon as he was released, he repudiated the treaty leaving his sons in a rather poor position. (They were released about four years later but were held in poor conditions.)

Why didn't this lead to war? In most of the examples, they were already in war. The only exception is Richard I and, even then, Philip II of France was involved (and Richard was to be at war with France on and off for the rest of his life.) This was part of war. You might get captured and then you'd have to pay. It was part of the rules. Francis I, as we are leaving the medieval period, had what we'd consider a more modern view of the ordeal. He didn't think anything he agreed to under duress counted and despised Charles V for the rest of his life.