It matters for filing taxes or filling out any forms that ask if you are married. You can’t say yes if the state doesn’t recognize your marriage. It is also important if you decide to get divorced from your spouse.
A legal marriage matters for inheriting property. If your spouse dies without a will, you automatically inherit their belongings and property. If you are not married you might not have any right to the property, even if you were with that person a long time.
Custody of children can depend on if the parents were legally married when the children were born. In Minnesota, if the parents are not married when a child is born, only the birth mother has custody of the child unless a court order says differently.
If you are not sure you are legally married under the laws of Minnesota, talk to a lawyer before filing taxes or filling out forms that list you as married.