Long answer here117. But in a nutshell both, yes and no.
Product Management can trace it's origins back to 1931! But what it was even 20-30 years ago is vastly different from what it is today.
Over the last couple of decades, we've been disrupted by the speed of technological advancement which left those who were once Product Managers managing physical products finding their skills being applied to managing digital products that could be updated and released to their customers daily rather than a new version coming out every 12 months.
As such the way we built products and did software needed to change - enter Scrum in the mid 90's.
Jeff Sutherland and Ken Schwaber the founders of Scrum came up with a solution to this problem. A framework where software would be built in an iterative fashion and could keep up with the pace of change and adaptability that was needed at the time. In doing so they created two new roles, one of which was titled the 'Product Owner'.
At that time Product Management was still evolving and for many they were still in the old waterfall/traditional Product Management mode of operating (agile hadn't been created yet). As such Ken and Jeff wanted this role to be different, they wanted to give it more responsibility and behave differently than how Product Managers were behaving at the time, hence they wanted to call it something else:
“When I created the Product Owner role, I gave it more responsibility for product strategy and revenue generation than a Product Manager. I specifically pulled the best Product Manager Easel Corporation had out of Product Marketing and retrained him. It also has more responsibility for directly working with engineering 50% of the time to assure that the product fits customer needs. So it is a broader role in that sense but usually does not include Product Marketing (sales collateral, shows) or long term Competitive Analysis although it needs to support those efforts.
The goal was to eliminate the common Product Manager failing of throwing requirements over the wall only to have the customer receive something that they didn’t want.” — Jeff Sutherland67
Ironically the Product Manager role itself, separately from what was happening with Scrum and agile, identified this problem too and begun to reshape the role to solve it.
No surprise that they too adopted working in more iterative fashion with closer collaboration with the development team and today the Product Manager role looks very much like what the original intent of the PO role was.
However whatever reason these two worlds, really trying to do the same thing, moved independently from each other. Agile found itself, rightly or wrongly shoehorned into software development for a long time whilst Product Management continued to be more on the business side.
Disconnected they begun to move in different paths and over the years the perception of the PO role narrowed whilst the PM role broadened.
This was further reinforced by frameworks like SAFe104 who claim that the PO role is an internal-facing tactical one, whereas the PM role is market-facing and strategic.
Today the definitions are so entangled it's hard to really give a straight answer. But the original intent was there, and arguably if you stick to that, the roles are the same.
Product Owner = Product Manager, they are the same thing but as market perception grows they are in many ways drifting apart. Much like the dictionary definition of words, their meaning and connotation have shifted over time. 117
However with the baggage of Scrum and agile having its software history and the connotations that steam from the like of SAFe, etc whom have turn the PO role into nothing more than a 'backlog-monkey', many prefer the title Product Manager.
For me, it doesn't matter as long as you don't fall into the trap of having both and turning one into a tactical 'backlog-monkey' and the other into a "strategic" role.