Thank you for asking about this. It's a pretty interesting subject.

To start with, there are two styles of fosterage:

1. Intra-household – The child stays in the parent's household and is "fostered" by a servant

2. Inter-household – The child is sent to another household to be raised

Let's talk about the intra-household fosterage first, partially because it is simpler and partially because it appears to be the style being referred to in the section of Gisla Saga you bring up. Now, as I say above, in the case of the intra-household fosterage the child remains with their parents and a servant or servants are chosen to serve in the role of foster parent. A foster-mother would have most likely been a wet-nurse, though would have remained as a motherly figure after the child was too old for nursing. A foster-father likely would have served as a bodyguard for a female child or as an instructor of some kind to a male child. So in the passage from Gisla Saga, the thrall described as foster-father would likely have been acting in the capacity of a bodyguard (or at least would have been earlier in the woman's life).

In the case of inter-household fostering there are three different subsets:

1. The fosterer is socially/financially inferior to the parents

2. The fosterer is socially/financially equal to the parents

3. The fosterer is socially/financially superior to the parents

In Icelandic law at the time, there was an obligation for members of a family to take care of those branches of the family that are not as prosperous. This meant that in many cases a well-to-do branch of a family would be forced to foster the children of poorer family members. This is mainly what the third case is referring to.

The other two cases would have involved non-kindred in most circumstances (case 2 was sometimes an arrangement between kin though). The main purpose of the first case was to solidify bonds of patronage. In exchange for the inferior family fostering the child, the superior family would provide some sort of boon. This could be in the form of providing additional livestock and farmland or guaranteeing to help the fosterer if they ever ran into trouble with their neighbors or other things along those lines.

The second case was used to patch up strained relations or to confirm legal settlements. There is a case in the Sturla Saga where a man sells some wormy meal to a son of Sturla. The man is then given the choice of being taken to "court" or of fostering one of Sturla's other sons. So the agreement to foster the child is used as recompense for ripping off the first son. Another case from I believe Njal's Saga is that an agreement is made where one son will marry Njal's daughter and another son will be fostered by Njal. This serves as a way to further solidify the bonds between the families.

Another aspect that plays into the fosterages is that it could serve as a way for the child to get an education that they would not otherwise get. This would usually only be a real concern in the second case of inter-household fosterage. There is at least one case in the sagas where a troublesome boy is sent to be fostered by another family with the dual purpose of having the boy learn to be better behaved, and also receive training in law (the foster-father was a well-respected law figure).

Not surprisingly, there might be a number of reasons motivating any particular fostering. Considerations of support and property were supplemented by concerns for education and training. Some fosterings were arranged to preserve peace in the parental household as in those instances where fathers sent away young unruly sons. We also see pubescent girls removed from the parental home to keep them safe from the attentions of young males. (Miller 124)

In terms of the kinds of relationships they had, it likely varied quite a bit. From the sagas it appears that foster-children were treated no different from any other children the family may already have had. Miller brings up a case where it is remarked with surprise that a foster-father does not require his foster-children to do normal chores, so it at least was the case that in general foster-children were not overly indulged. Also, even though the fosterages were used as a form of dispute settlement and could thus be seen as a form of hostage taking, there is no evidence in the sagas that they were ever treated as such. Again, as far as the information that has survived, it appears that regardless of the circumstance behind the fosterage the children were treated like every other child.

Last thing has to do with the legal status of foster family:

With both types of fostering [intra- and inter-household], the sagas consistently portray the bond running as, if not deeper than, that between parent and child. The laws also recognize the strength of this bond. A man has the privilege to kill for sexual assaults on his foster-daughter and foster-mother, who are thus members of a class that includes his wife, daughter, mother, and sister. (Miller 171)

There are plenty of cases throughout the sagas of foster-children carrying out vengeance for the deaths of their foster-parents (though Miller does point out that many of those cases involve children being fostered by people who are kin, so make of that what you will).

Hopefully that was enough to sate your curiosity, but if you end up with any other questions just let me know and I'll see what I can find.

Source:

Miller, William Ian. Bloodtaking and Peacemaking: Feud, Law, and Society in Saga Iceland. Chicago: University of Chicago Press, 1990.