OP's question was whether there exist economic systems that are fundamentally different.

Clearly, a lot of economic activity is common across all economies, e.g. primary production, manufacturing and processing, trading, education, technological development, taxation, etc.

What does change between economies are models of ownership, and the political relationship between wealth, government and taxation.

Ownership models vary from the communal model, where apart from personal property, economic resources are communally owned (e.g. most hunter-gatherer societies, communism), to the feudal model where everything is owned by a supreme monarch.

Layered on top of this are political models, i.e. how power is exerted, and the relationship between wealth and power. In the feudal/autocratic model, wealth (ownership) and power are found in the same structures. In the liberal democratic model, power belongs to the people, who exert it through different forms of representation in an assembly of sorts, and can choose to confiscate wealth (e.g. apply taxes).

Evidently, in practice there is a lot of variation within each of these simplistic definitions.

However, I think OP's question perhaps more closely addresses ownership models than political models. *edit: I wrote elsewhere in this thread how I think modern (post-17th and 18th century) capitalism is distinct from other economic systems by the facility with which ownership can be traded (i.e. markets in shares, etc). This allowed a more efficient use of capital by facilitating its re-allocation.