In Sweden, a Master in Social Work may be your best bet - or perhaps a joint MBA + Master in Social Work (which is more common that an MBA for Non-Profits specifically).
At the moment, our MBAs in Sweden are centered on that general MBA. However, Wharton Business School has an excellent joint MBA + Master in Social Work, which you can look into. Alternatively, University of Liverpool (and many others) offer an online Master in Social Work, which can be taken from anywhere.
MBAs specifically for non-profits do exist and usually have titles a bit like 'MBA in Non-Profit Management' - they are just not the most common option. Professionals working in the non-profit industry may seek a general MBA or an EMBA well into their careers.
Non-profit organizations that do not belong to the government are not so common in Sweden as the welfare system covers many of the services that non-profit organizations in other countries provide. The few non-profit, non-governmental organization in Sweden are large, international bodies such as Green Peace and the Red Cross.