Mutuality is a fuzzy concept. Mutuality is not virtuous in itself. It depends who benefits. Some mutual organisations act against the public interest. The main purpose of BUPA is to enable people to jump the queue for hospital treatment.
Mutualism is a fuzzier concept than socialism.
However there has always been a strong link between socialism and mutual aid. Self help and mutual aid are the two main cornerstones of the trade union movement. Mutuals should be independent of both the state and private financial power. They can be bulwarks against over-powerful government and the abuse of market power by big business. In theory, because they are democratically controlled, they should enable ordinary people to have a say in the way their money is used.