The answers to those questions depend heavily on your company/institution/hierarchy. I think we can only respond in the context of PMI (although I'd love to hear from a PRINCE practitioner or other non-PMI organization).
Work performed as a project coordinator almost certainly qualifies as job experience towards PMP certification (I'm only hedging because I'm not accountable for making that determination). In that sense, work as a project coordinator does bring you closer to Project Manager title. However the meaning of those titles depends on whether your organization is weak matrix, strong matrix, or some combination, the internal politics and history of your institution etc.
Ultimately, the title of Project Manager is awarded by your company/institution. Both my manager and I are PMP, but he is a project manager and I'm either a project coordinator or a senior cybersecurity principal analyst. On the other hand both he and his colleague are Project Managers, despite the fact that they both work on closely coordinated programs (technically not projects, but politically we can't call them programs). In our case the title is determined by politics.
To put the same idea a third way, I think your last question is the most pertinent "how do they compare in the management hierarchy?", but that question is entirely localized. "Management Hierarchy" is a term that has no meaning beyond the scope of YOUR management hierarchy. Only your management can determine whether your performance as a project coordinator qualifies you for consideration as a project manager.