What would be a good job title of the person who is typically tasked to run and manage a PMO?
Good Job Titles Describe Officeholders' Roles
I'm looking for a job title as an answer in both questions.
The underlying problem is that you're assuming there's a canonical answer to your questions about organizational structure and job titles. There isn't.
Instead, you should focus on describing the role. According to Wikipedia:
The PMO strives to standardize and introduce economies of repetition in the execution of projects.
What this actually means within the context of any given organization will vary. You can spoon around titles as you please. The important thing about titles isn't what you call something; what matters is that the title communicates effectively about the named role. As long as the titles are meaningful within your organization, and serve as a useful shorthand to describe the scope of the PMO's services within your company, that's all that matters.
Having said all that, if your PMO is a one-man operation, the title of "PMO Facilitator" suggests itself. You are essentially describing a process coach or facilitator role, but one with very limited authority. Calling this person a facilitator clearly communicates the scope of the role without implying an executive or managerial function that hasn't been formally delegated.
Your best bet is to meet with the stakeholders to whom the PMO reports, and ask them to describe the role of the PMO as they see it. If all the stakeholders can agree on what the role is, they can then collectively agree on the appropriate title to describe that role succinctly.