I guess? Mostly the cool applications in physics and differential geometry are about tensor fields, which are more complicated than bare tensors. You could argue that they're talking about finite dimensional tensors but tensor fields are kinda a different object (at least subjectively, to me).
A tensor field is a finite dimensional object varying over space. The space of all tensor fields is infinite dimensional. An operator in QM is infinite dimensional at a single point itself, and in QFT we have fields of such operators.
