There are lots of different kinds of "isomorphism," depending on what category we're talking about. When the line of real numbers and 3D space are both regarded as smooth manifolds in the usual way, they are not isomorphic to each other as smooth manifolds. In other words, regarded as objects in the category of smooth manifolds, they are not isomorphic to each other. In that category, "isomorphism" means "diffeomorphism." They're not isomorphic to each other in the category of topological spaces, either, at least not when they are equipped with their usual topologies. In that category, "isomorphism" means "homeomorphism."
When people count degrees of freedom, they're doing that counting in the context of whatever mathematical structures are important for the model's definition — which usually includes at least a topological structure so that continuity can be defined, and usually includes a smooth structure so that derivatives can be defined.