There's actually a pretty comprehensive answer to this question, which I'm excited to discuss. I'm not an expert on monastic institutions per se and I welcome anyone more knowledgeable correcting or nuancing this response, but basically, I'd like to approach your question by talking about child oblation in medieval Europe and contrasting it with other forms of religious initiation.

(Quick caveat: You say that your question is inspired by something you read about Greece; I can't really speak to the Greek side of the question, but will be focusing on the European middle ages. If you were looking for a specifically Byzantine answer to this question then I apologize.)

The most archetypical monasticism in the Latin Christian world is the Benedictine tradition. While Benedict of Nursia (c. 480-550) was not the first western monk, his monastic code, The Rule of St. Benedict, laid the groundwork for monastic life in communities ordered under an abbot. The Rule, Mayke de Jong argues, was not the first place we see children inserted into the monastic life, but it was innovative in that it established the initiation of a child (puer) into the monastic community within the context of a Eucharistic celebration (the Mass). This is what Benedict's Rule says about young oblates - that is, children offered to a monastery:

If a noble person offers his son to God in the monastery, if the child is still young, let his parents draw up the document we have already discussed, and enfold the hand of the boy in the altar-cloth together with the offering and offer him in this manner.

As far as their possessions are concerned, the document in question should contain their solemn oath never to give him anything nor to provide him with the opportunity to possess anything - either themselves, or through an intermediary, or in any other way; if they do not wish to do this but rather, for their soul's sake, would like to give alms to the monastery, let them make an offering of which - if they so choose - they may retain the usufruct. Thus let all roads back be cut off, so that the boy has nothing further to look forward to that might, as we know from experience, tempt him and - God preserve us! -lead him to damnation.

Let poorer people do likewise. But those who really have no possessions at all should simply draw up the document and offer their son with the offering in the presence of witnesses.

The whole business about isolating the child from property and the outside world may seem pretty sinister to contemporary listeners, but there's a rationale here - Benedict is concerned that children who enter a monastery, like any other monk, sever ties with the profane world and live the ascetic life. There's much more to be said about child oblation in Benedict's day, but with that little crash course out of the way, we should consider the development of the practice over time. In the Carolingian period, the majority of monks came to their communities via oblation: "Nutriti [monks raised in the monastery] dominated the scene, while adult novices often ended up in monasteries only as the result of some sort of punishment, be it a penance or political imprisonment" (130). De Jong posits that, among other reasons, this is linked to the need for monastic institutions to need increasingly large numbers of clerics to offer increasingly large numbers of Masses, as well as the desire for monks to be sexually pure, and thus an incentive to recruit monks who had not yet reached sexual maturity. Lastly, in a monastic world increasingly stratified by education, those who entered a monastery young had more time to become steeped in texts, whereas an adult conversus might struggle to learn even basic reading.

Again, this is only a broad sketch, but I want to take us out of the Carolingian world and to England, where I'll be drawing on a great dissertation written by Rebecca King Cerling. As Cerling explains, offering a child was not only a way for nobles to avoid the problem of a landless younger son, but to create a relationship with an institution and ensure prayers and Masses for their salvation and that of their relatives. Child oblation was not the only way that families could associate themselves with monasteries - Cerling also writes about monastic confraternities - but the point here is that families offered children not only out of self-interested desire for prayers on their behalf, but as part of broad strategies to strengthen the social ties between themselves and monasteries. Child oblation here is symbolic - sacramental, even, if I can be cheeky - of the relationship between monasteries and their surrounding communities. Here, again, monks who entered the institution as children were likely to become its most educated members - Cerling tells us that cantors, liturgical leaders in the community, were usually child oblates, because they had accumulated the greatest experience and mental inventory. This holds true for monastic women as well, by the way - while most of the reading that I've done has focused on men, I can't pass up an opportunity to mention Hildegard von Bingen, by favorite medieval saint. Before going on to become a brilliant polymath (composer, author, visionary, wonderworker, abbess), the child Hildegard was, you guessed it, given to a hermit woman named Jutta, who taught her to read, setting her on the path for a brilliant religious career. I think that this helps to emphasize that even though oblation used children as currency in an exchange between a religious institution and a family, these children often went on to achieve success, learning, and status unavailable to most of their peers.

However, I don't want to give the idea that child oblation was a universally-accepted practice. Not only, as de Jong tells us, was it resisted in the early days of The Rule of St. Benedict, but later monastic reformers had little use for it either. The Cistercian order, while it accepted boys for training in music and academic learning, would not accept formal oblates until adults (sixteen-year-olds, in this case), and these were meant to come to the community of their own volition, rather than their parents'. Child initiation is also one of the areas in which the mendicants of the later middle ages (e.g. Franciscan and Dominican friars) distinguished themselves against their monastic counterparts. To illustrate this, we can turn to the famous story of Thomas Aquinas. The son of minor nobles, Thomas' father meant for him to enter the monastery of Montecassino, where one day he would become abbot, obviously a prestigious prize for his family, but Thomas instead resolved to enter the fledgling Order of Preachers (the Dominicans), sparking a conflict with his family punctuated by such episodes as Thomas' brothers hiring a prostitute to sabotage his piety, only for Thomas to chase her away with a burning log plucked out of a fireplace. Aquinas, of course, went on to have a brilliant academic career of his own, becoming one of the great scholars of the University of Paris in addition to achieving sanctity in the Dominican model. I think that this illustrates for us not only how ingrained were the assumptions about the relationship between nobles and monasteries, but also how the appearance of mendicant orders subverted those assumptions and offered alternatives to them.

This is a long post, but let me give some general points:

I. Child oblation had a long history with Benedictine monasticism, which envisioned it as an agreement between the child's parents and the monastery.

II. This was for much of the middle ages the primary source of new monastics, and child oblates often achieved high positions in monasteries due to their long exposure to monastic learning.

III. Resistance to the practice was associated with (some of) the religious reforms of the later middle ages, both those that wished to reform the Benedictine tradition (e.g. Cistercians) and those that tread new ground (e.g. mendicants).