In introductory textbooks (Griffiths, Shankar, Boas) a tensor is introduced as a mathematical objects which transform in a specific manner under changes of basis (i.e. changes of the coordinate system). Specifically, a 0-rank tensor (also called scalars) is independent of such changes, whereas any other $n$-rank tensor ($n \neq 0$) is not. Exceptions to this rule are the class of isotropic tensors (can however be of arbitrary rank), whose components are the same regardless of the choice/change in the coordinate system.
If isotropic tensors do not need to undergo any transformation when changing the coordinate system, why are they thereby not considered to be scalars (since the whole concept of a tensor of rank $n \neq 0$ is based on the concept of obeying certain tranformation rules)?
What am I missing here? (I know that by the quotient rule, the rank of these isotropic tensors can be confirmed to be non-zero.)