I am not exactly sure if this is the sort of answer you are looking for, but here it goes. It seems that the actual question you are asking is about the unstable comparison of homotopy fiber and cofiber, and I am not convinced that working in spectra really solves the problem.

The classical examples concerning the interplay of homotopy fiber and homotopy cofiber come from the loop space fibration resp. the suspension cofibration. For a space $X$ we can consider the loop space fibration $\Omega X\to \ast\to X$, and then the cofiber of $\Omega X\to \ast$ is $\Sigma\Omega X$. If $X$ is a group-like $H$-space, this may be an example of your situation. Similarly, the homotopy fiber of $\ast \to \Sigma X$ is $\Omega\Sigma X$. The precise relation between $X$ and $\Omega\Sigma X$ is given by the Freudenthal suspension theorem - you get isomorphisms on homotopy in some range, and outside that range you might still be able to say something using the James model.

More generally, you may be able to use the relative Hurewicz theorem to get a relation between the homotopy fiber (controlling the relative homotopy groups) and the homotopy fiber (controlling the relative homology groups). You might want to have a look at the discussion of the relative Hurewicz theorem in the "Simplicial homotopy theory" book by Goerss and Jardine.

Finally, I am not sure if I would agree to a statement like "for spectra cofibrations and fibrations are the same". They are still different classes of maps in the model structure. Certainly cofiber sequences and fiber sequences are the same.

I am not exactly sure if this is the sort of answer you are looking for, but here it goes. It seems that the actual question you are asking is about the unstable comparison of homotopy fiber and cofiber, and I am not convinced that working in spectra really solves the problem.

The classical examples concerning the interplay of homotopy fiber and homotopy cofiber come from the loop space fibration resp. the suspension cofibration. For a space $X$ we can consider the loop space fibration $\Omega X\to \ast\to X$, and then the cofiber of $\Omega X\to \ast$ is $\Sigma\Omega X$. If $X$ is a group-like $H$-space, this may be an example of your situation. Similarly, the homotopy fiber of $\ast \to \Sigma X$ is $\Omega\Sigma X$. The precise relation between $X$ and $\Omega\Sigma X$ is given by the Freudenthal suspension theorem - you get isomorphisms on homotopy in some range, and outside that range you might still be able to say something using the James model.

More generally, you may be able to use the relative Hurewicz theorem to get a relation between the homotopy fiber (controlling the relative homotopy groups) and the homotopy cofiber (controlling the relative homology groups). You might want to have a look at the discussion of the relative Hurewicz theorem in the "Simplicial homotopy theory" book by Goerss and Jardine.

Finally, I am not sure if I would agree to a statement like "for spectra cofibrations and fibrations are the same". They are still different classes of maps in the model structure. Certainly cofiber sequences and fiber sequences are the same.

I am not exactly sure if this is the sort of answer you are looking for, but here it goes. It seems that the actual question you are asking is about the unstable comparison of homotopy fiber and cofiber, and I am not convinced that working in spectra really solves the problem.

The classical examples concerning the interplay of homotopy fiber and homotopy cofiber come from the loop space fibration resp. the suspension cofibration. For a space $X$ we can consider the loop space fibration $\Omega X\to \ast\to X$, and then the cofiber of $\Omega X\to \ast$ is $\Sigma\Omega X$. If $X$ is a group-like $H$-space, this may be an example of your situation. Similarly, the homotopy fiber of $\ast \to \Sigma X$ is $\Omega\Sigma X$. The precise relation between $X$ and $\Omega\Sigma X$ is given by the Freudenthal suspension theorem - you get isomorphisms on homotopy in some range, and outside that range you might still be able to say something using the James model.

More generally, you may be able to use the relative Hurewicz theorem to get a relation between the homotopy fiber (controlling the relative homotopy groups) and the homotopy fiber (controlling the relative homology groups).

Finally, I am not sure if I would agree to a statement like "for spectra cofibrations and fibrations are the same". They are still different classes of maps in the model structure. Certainly cofiber sequences and fiber sequences are the same.

I am not exactly sure if this is the sort of answer you are looking for, but here it goes. It seems that the actual question you are asking is about the unstable comparison of homotopy fiber and cofiber, and I am not convinced that working in spectra really solves the problem.

The classical examples concerning the interplay of homotopy fiber and homotopy cofiber come from the loop space fibration resp. the suspension cofibration. For a space $X$ we can consider the loop space fibration $\Omega X\to \ast\to X$, and then the cofiber of $\Omega X\to \ast$ is $\Sigma\Omega X$. If $X$ is a group-like $H$-space, this may be an example of your situation. Similarly, the homotopy fiber of $\ast \to \Sigma X$ is $\Omega\Sigma X$. The precise relation between $X$ and $\Omega\Sigma X$ is given by the Freudenthal suspension theorem - you get isomorphisms on homotopy in some range, and outside that range you might still be able to say something using the James model.

More generally, you may be able to use the relative Hurewicz theorem to get a relation between the homotopy fiber (controlling the relative homotopy groups) and the homotopy fiber (controlling the relative homology groups). You might want to have a look at the discussion of the relative Hurewicz theorem in the "Simplicial homotopy theory" book by Goerss and Jardine.

Finally, I am not sure if I would agree to a statement like "for spectra cofibrations and fibrations are the same". They are still different classes of maps in the model structure. Certainly cofiber sequences and fiber sequences are the same.