Does Denmark not also have an official church? I'm not sure about the other Scandinavian kingdoms.
The (Lutheran) Church of Denmark, as far as I understand, is an established church in the sense that the Queen is the supreme authority and church ordinances have to be adopted by Parliament ( although I was told that, in practice, such ordinances are never changed). But there is no coronation or annointment of the monarch (at least not anymore).
The Church of Sweden was separated from the state, meaning that the church is now completely self-governed through its General Synod and its archbishops, bishops and parishes. Certain elements of the church constitution, however, including the obligation that the church must follow a Lutheran doctrine and be organized in parishes and dioceses, are defined in an act of the Swedish Parliament, which, I believe, can be changed only by Parliament itself.
Similarly, the Church of Norway is also self-governed now, but retains a constitutional status of "national church" and receives public funding.
Both in Denmark, in Norway and in Sweden, the monarch
must be a member of the respective national Lutheran church. And, in Sweden, in addition, all "princes and princesses of the Royal House" must be raised in the Lutheran faith adopted by the Church of Sweden and "members of the Royal Family" (I am quoting the Act of Succession) who do not profess that faith are excluded from the line of succession.
So my impression as an outside observer is that technically Denmark is the only country with an established church in the sense of a church that is under state government through the civilian authorities, but the churches in Norway and Sweden are still "semiofficial" in the sense that they are partly regulated by public law and the monarch must be a member of the church.