Right, markets are an institution but not the only one. It's a problem that people forget and think it is only one we need.
Economic institutions encompass everything that allow it to function including basic ideas like private property rights, rules and regulations that enforce contracts, and functions that account for the welfare of people.
Markets are what people with diverse motivations and rights to dispose of their property "do" when they wish to dispose of their property (or deliver services) or when they wish to acquire new property or have services performed. The market as a concept isn't something handed down by the gods, nor instituted by decree.
I agree that there are other concepts that dispose of people's "property" and require them to seek or deliver services; in the past we called them things like theft, war, and slavery.
Economic institutions encompass everything that allow it to function including basic ideas like private property rights, rules and regulations that enforce contracts, and functions that account for the welfare of people.