TL; DR: Uncommon, but roughly based on the basic connotation of the fostering practice in Old Norse texts.
In Old Norse-Icelandic sagas, the two families sometimes make use of the social practices of fostering (as well as concubinage) to build the social bond between the parties involved (Miller 1990: 122-24; Jón Viðar Sigurðsson 2007: 84f.). Note that the nature of bond built on these two kind of social relation essential implied inequality while both parties gained some profits in reality: In the most commonplace type of the fostering, to be foster parents were usually of lower social standing than who sent their child to fostering. It's true that there were some cases between those who of equal social ranks, such as between two Icelandic chieftain families, but in most cases the parties involved had already been of good terms like kinsmen by blood or by marriage (Miller 1990: 123).
In turn, to accept to be a foster family could mean to accept their social inferiority, at least in some Old Norse text: The closest parallel of OP is found in the episode of fostering of King Hákon the Good, future king of Norway (d. 961) as well as son of King Harald Fairhair of Norway(d. 932/3?) (Saga of Harald Fairhair, Caps. 38f., in: Hollander 1964: 92f.). At first, King Aethelstan (d. 939) of Wessex sent a sword to King Harald as a sign of mockery: King Harald realized that a sword symbolized an overlordship of King Aethelstan also in Norway. Then, in return, King Harald commissioned his young (and illegitimate) son Hákon to Hauk, one of his retinue to take him to the court of King Aethelstan in England. At the court of Aethelstan, Hauk put young Hákon on King Aethelstan's knee without any notice, and then stated that: 'King Harald bade you foster for him the son of his maidservant' (Hollander 1964: 93). Thus, King Harald of Norway and Hauk pushed the inequal fostering of his young son by wit in requital of Athelstan's sword 'gift', at least in the saga. This episode does not concern the mutual fostering between two parties, as asked in OP, but at least the fostering was employed to make up of the similar inequal symbolism of the sword-gift.
References:
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Edit: I've just seen that u/y_sengaku has provided a fantastic answer on the actual fostering practice. This works out nicely, as this is something my answer largely neglects!
In A Song of Ice and Fire (ASOIAF) we're shown that it is a fairly common practice for the Lords of Westeros to raise the children of other houses after some form of agreement has been made between these two houses. Perhaps the best example of this is when shortly after the failure of the Greyjoy rebellion, Balon Greyjoy swears fealty to the Baratheons and gives his son, Theon, as a hostage to Eddard Stark. In ASOIAF Theon is essentially raised by Eddard, almost becoming a member of his family. However, was this practice reflective of peacemaking methods in the realworld within the middle ages? This is a complex question with different answers depending on where and when you focus on during the medieval period. I focus on treaties made by medieval peoples in Europe and the Mediterranean (specifically I compare treaties made by the Kingdom of England and the Byzantine Empire from 900-1200 C.E.) so I will focus on this area of the world.
Using hostages was a staple of peacemaking throughout the medieval world of the Mediterranean and Europe. While there were certainly other methods available to medieval peoples, such as enforcing an agreement by making the two rulers and peoples involved swear oaths (often of a religious nature), it was always difficult to ensure that an agreement was kept in the future. For example, the Treaty of Devol, made between the Byzantine Emperor Alexios I and the Normano-Sicilian Prince of Antioch Bohemund in 1108, has Bohemund swear:
"by the passion of Christ, our passionless Saviour, and by his unconquerable Cross, which He endured for the salvation of all men, and on these all-holy Gospels, which lie before me, which have brought the whole world into their net; holding them in my hand, and in my spirit, I include with them the most precious cross of Christ and His crown of thorns and the nails and the lance which pierced his divine and life-giving side..."
This treaty essentially incorporated the Principality of Antioch into the Byzantine sphere of influence. It was made after Bohemund had waged a disastrous campaign against Byzantium, where his army had been ravaged by disease, essentially making Bohemund's negotiating position very weak. Unfortunately for Emperor Alexios, it was never actually enforced, as Bohemund died before he could return to Antioch, and his nephew Tancred, Bohemund's heir, refused to honour the treaty. This in essence sums up the problems facing medieval rulers when making agreements with another people; how does one ensure that an agreement continues to be followed once the circumstances that produced the original agreement change? How was Alexios to ensure that this agreement not only bound Bohemund once Bohemund had escaped the precarious situation that the the Namanno-Sicilian ruler had found himself in prior to the treaty, and furthermore, how could Alexios ensure Bohemund's successors accepted an agreement that they themselves did not make? It is worth noting here that this is not a problem that has 'gone away' and is still a concern to modern nation-states! Regardless, the point in the above example is that even though Bohemund swore a sacred oath, upon the gospels and other holy relics, the treaty was still un-observed. This is where other methods or enforcing a treaty come in, such as hostages.
The example of Balon Greyjor in ASOIAF giving his heir as a hostage implies that Robert Baratheon had enough foresight to realise that any agreement the two rulers made could be broken in the future. By taking Balon's heir hostage, he is not only excerting pressure to abide the agreement over Balon, he's also safeguarding himself against a Greyjoy successor, likely being either Euron himself, from breaking the agreement. Here, it is worth noting the Ironborn have an unusual way of nominating their rulers, being the Kingsmoot. However, while I am certainly no expert on ASOIAF, it is worth noting that these meetings generally seem to nominate the son of the previous king (not that this is always the case). Even with the difference in the inheritance of the Lordship/Kingship of the Iron Islands, it is clear that Robert taking Theon hostage makes it more likely that both the current and future Lord of the Iron Islands would observe the agreement.
This certainly has historical parallels. What initially came to my mind when reading your question was the chronicle accounts of medieval Irish history. In 1167 Diarmait Mac Murchada, the King of Leinster, was deposed by the then High King of Ireland, Ruaidrí Ua Conchobair. Diarmait was forced to flee Ireland, and search abroad for aid. He returned with English and Anglo-Welsh allies, in the form of 'Strongbow', Earl of Pembroke. Diarmait subsequently recovered much of his land, and eventually was confronted by Ruaidrí in 1170, where Ruaidrí recognised Diarmait as King of Leinster in return for Diarmait's submission, and Diarmait giving his son as a hostage to Ruaidrí. While one might think this assured Ruaidrí of Diarmait's desire to keep the peace, this doesn't seem to have worked, as Diarmait broke the peace soon after, and presumably his son was killed, his function as a hostage no longer guaranteeing the peace.
While there are certainly hostages in many treaties during this period, they are not always the children of other lords. For example, the 1174 Treaty of Falaise between King Henry II of England and King William the Lion of Scotland has William give his brother David, along with several other important lords, as hostages. However, this treaty does not only rely on hostages as sureties of the treaty. It also states that:
"By way of security for the strict observance of the said covenants and final agreement so made with [Henry]... on part of the king of Scotland and his heirs, the king of Scotland has delivered unto [Henry] the castle of Roxburgh, the castle of Berwick, the castle of Geddewerde, the castle of the Maidens, and the of Striveline..."
This treaty was clearly actually observed, as it is later undone by Henry's successor, Richard, at the Treaty of Canterbury, in return for funds to go on the Third Crusade.
I realise that so far this answer hasn't really commented on the actual fostering of hostages that were the children of a rival party (as seems to be the case of Euron in ASOIAF). This is actually very hard for me to comment on. Certainly, while there are hostages in treaties, it is rarely noted how these hostages were to be treated. This isn't to say that the fostering of hostages did not happen, but rather, as far as I'm aware it did not occur in any English or Byzantine treaties from 900-1200. I suspect that if this did occur, it is likely these conditions were not agreed via treaty, but rather outside of the treaty agreement. This makes ASOIAF more interesting in some ways, as, I don't believe any of the books actually state Euron being made a foster child Eddard is due to an actual named treaty? I am not expert on ASOIAF though. Regardless, I hope this helped answer your question a little!
Sources:
I've tried to use sources that are freely available online and are available in English, so please, have a read of the sources if you'd like to learn more!
The Treaty of Devol is available in Anna Komnene's Alexiad, and is one of the few Byzantine copies of a treaty from 900-1200. You can read it here (pages 247-253).
The Anglo-Norman invasion of Ireland and the subsequent events described are found in Gerald of Wales' The Conquest of Ireland. You can read the events I refer to here (pages 12-21).
The Treaty of Falaise and the Treaty of Canterbury are found in the Chronica Majora of Roger of Howden. You can read the Treaty of Falaise here (pages 398-401) and the Treaty of Canterbury (referred to as a charter in this edition) here (pages 131-132).