A CEO of a company is ultimately responsible for setting the overall vision, and making sure the team has the right talent and organizational structure to accomplish that vision and enough funding in order to achieve it.
The “product CEO”, especially one of an early stage company may spend time doubling as a product manager and be very hands on with helping the team to build the product. They will clear roadblocks, make product decisions, etc. Even as a company scales, the best product CEOs are able to keep pushing the vision and expressing that with teams at fairly detailed levels with feedback and pushing their teams. Though you don’t really want them involved in making every micro decision.
As a company scales and adds full time product managers — those PMs rarely if ever get to decide on funding, corporate strategy, nor hiring and firing overall resources for their broader team. Those decisions are generally made at the executive levels. This is why I don’t like saying a product manager is a CEO.
That said — there are a lot of CEO-like responsibilities and traits that a product manager does need to take on. For example: accountability for what the team delivers, articulating and reinforcing the vision, and flexibility to do whatever it takes to help.
This is is why many of the best product managers go on to become founder/CEOs of companies and why I love to fund and work with them too.