Your question is absolutely valid and also important. In fact, most of what I write in my paper and the blog is about answering precisely this.

My observer is well defined, as a mechanism that is part of a quantum system and who interacts with the quantum system to gather information about it. He is limited by the locality of interaction and the unitary nature of the evolution. I imagine the observer to be a physicist, who tries to describe the universe mathematically, based on what he sees. But that is only a trick in order to have a mathematical formulation of the subjective view. The observer is prototypical for any mechanism that tries to create a model of his surrounding. This approach is very different from modeling cognitive mechanisms, and it's also much more general. The information restriction is so fundamental that you can talk about his subjective reconstruction of what is going on as local subjective reality, as everyone has to share it.

The meaning of the dominant eigensubspace is then derived from this assumption. Specifically, I am able to identify a non-trivial transformation on the objective density operator of the observer's subsystem that he cannot gain any knowledge about. This transformation creates a class of equivalent representations that are all equally valid descriptions which the observer could use for making a model of his environment (and himself). The arbitrariness of the representation connected with this reconstruction however forces him to reduce his state description to something more elementary, something that all equivalent descriptions have in common. And that turns out to be the dominant eigensubspace as his best option. This point is very important, and the derivation I provide in the blog is rigorous and detailed. The result is that the subjective reality as reconstructed by any observer like this evolves unitarily if the greatest eigenvalue does not intersect with other eigenvalues (the observer himself cannot know the value of the eigenvalues either) or discontinuous as a formerly smaller eigenvalue intersects with the greatest one to become the new dominant eigenvalue. This requires an interaction with a part of the system that is not contained in the objective local state description, like an incoming photon.

This approach also has the advantage that you don't have to actually model the observer. You still know what information is available to him. That is why the observer does not even have to be part of the system that you want to "subjectify". You already know how he would describe it. Specifically, you don't have to consider any kind of entanglement between observer states and observed states. The dominant eigensubspace is a valid description of every system that the describing entity is part of and that contains everything the observer is directly interacting with. If you want to get quantum jumps you also need an external inaccessible environment.

Summarizing, there's no need to postulate the ontology or relevance of the dominant eigensubspace. I was very careful to only make assumptions that are transparent and to derive everything from there. Specifically I am not adopting any definition or terminology from interpretations of quantum theory.