It's a weird arrangement. And yeah, I'm not actually sure myself. I think it's probably closest to how jetBlue did things prior to the arrival of ALPA. As far as what kind of an entity it is, I'm not a labor lawyer and it's best to not ask me.
I take "pilot protection" to be a very broad term - in the services sense, we actually do have (for instance) aeromedical services (the Company has agreed to pay for AMAS, for instance) and a handful of other professional services available, although nowhere near as much or as varied as the ALPA carrier I worked for. We also have some reasonably robust pilot assistance services, including HIMS, CIRP, and Professional Standards, and are starting up an additional program to allow pilots to better manage stress and trauma not solely related to their occupation (I think American calls this Wingman and Delta calls it Pan-Pan, although I'm not strictly sure). Pilots additionally have the right to have one of our representatives present during any management interaction, and we have a defined process for disciplinary actions and termination appeals as well.
In the "contract language" sense, it depends on the issue of the week. I view the Pilot Policy Manual as equivalent in force to a contract (at least in the sense of if I fail to follow it I will surely be fired), and expect that it is followed. I spend the bulk of my time as a rep answering questions and developing policy ideas, actually, and getting them into language. (Whether or not they go anywhere after that is another question.)
Clear as mud, no doubt.