Hi ScottishWalrus and welcome to the site!
I think you're referring to "holmgang", and it's definitely a great example of the danger of learning history from TV series or films. There are two things to bear in mind about it - first of all it seems to have had legal status only in one part of Sweden though similar settlements of dispute were recorded around Scandinavia on a more ad hoc basis, if the sagas reflect real incidents (they don't always). Secondly it wasn't so much about possessions as about "honour and shame".
Your ambition in Viking society was to achieve and hold on to "dnengr" status. For a man or a woman this was all about being honest, strong willed, and handy in a fight if needed, all of which earned you respect. The status had to be conferred on you, normally after a "tingmøte" (an official meeting of the community that considered how things had been going and planning for the coming season). The opposite to this was "nidingr", also conferred on you if you really pissed the others off and brought shame on yourself or - worse - on the community through deceit, cowardice or any other "slem" behaviour.
A "nidingr" was in real trouble. He (or she) was fair game to be challenged by anyone to a form of duel (no stabbing or slicing weapons normally allowed) which could result in them being exiled or indeed executed if they lost. If they were sole owners of possessions then these too could be forfeit, but it's worth remembering that other than weapons, clothes and a few utensils it was likely that possession was communal, or shared with a family or a spouse, so these were never the "prize" for winning a "holmgang". In one exceptional case, and we know this basically from one incident in one saga featuring pseudo-historical characters, a "nidingr" challenges a "dnengr" who in fact had cheated his way to that status and the result is indeed a total flip of status for both individuals, the loser then taking his own life. In reality a "nidingr" stood to gain little from winning such a fight - he might have clawed back a little respect and avoided exile, but a fast track form of social climbing it most definitely wasn't!
A wife, by the way, would have been completely off limits. She was no man's "possession" but had all the rights of a man (including unilaterally declared divorce after which she kept all her stuff plus whatever she reckoned was fair compensation from the ex), and had you thought you'd "won" her in a "holmgang" with her husband I reckon you'd quickly be finding yourself challenged to one yourself - by her. Though more probably, having insulted her status for no good reason, you'd more likely be told to skip the whole "holmgang" thing completely and simply consider yourself a mega-"nidingr" and "utkast" by the next "ting". Not a great place to be when the winter kicks in! Though I dare say you'd have her ex for company out in the icy wilderness.