I don't know if their commitment to a pious life stopped them from forming close friendships but it seems unlikely that a monk living in the same monastery for most his life (assuming thats what happened ) would not start to form friendship if not familial ties to his fellow monastic "brothers". Do we have any records of this? How did this relations affect day to day life? Did they have any outside relations as many monasteries were integrated to the surrounding communities? If the sources arnt available for my time frame, the question can open up to any time in history.
It would depend upon the other - mendicant, eremitical, preaching, self sufficient, etc. if there is a particular order I could tell you based on their rule, which all seem to somewhere have roots in the Benedictine Rule,