Yes. That's why it's best to say "observer dependent" or "observer independent". Which I tried to mention later on.
It's my own personal phillosophy (and not any generally agreed on statement) that says that observer independent quantities are 'real" and observer dependent quantities are not. I''d guess this is a common enough position to have a name in the philosophical literature, but I don't know what that name is, alas.
There are some other issues here as to how one might define observer dependent or observer independent. The individual components of tensor quantities vary between observers, for instance, but I'd regard the tensor as a whole as falling in the "observer independent" category.