You're begging the question. What does "engineering the product" mean? Software engineers are not civil engineers, there are not professional standards for what this means.
Fair point, there's no formal or widely accepted definition for Product Engineer.
In Roman languages, the word "engineer" derives from the word for "wit" (e.g. "Ingeniero" <- "Ingenio" in Spanish); hence there's a mental connection to identify an engineer as someone who uses her wit to solve problems. The Product Engineer hence uses her wit to solve Product issues, that could range from technical issues to manufacturing, materials, distribution or even marketing issues. She has to have a more holistic and less deep view of multiple areas, a la Project Manager.
Now you'll tell me that I'm talking about a Product Manager and I'll have to agree. In my opinion, a Product Engineer is a different name for a Product Manager. Product Management is a really broad area and I think some people more inclined to technical stuff and especially software development would feel more at home being called Product Engineer than Product Manager.
So here's my problem with this. There is no bricklaying in software. If something is really repetitive and mind-numbing it should be automated. That is the difference between software engineering and engineering/construction—the computer does all the rote work.
I get that there are amazing semi-technical managers and UX people who add outsize value even though they don't code—I don't subscribe to the SV worship of "hard skills". However, SV is also full of jackass wantrepreneurs who think they don't need to learn how to code because their singular vision is so valuable. These "idea guys" are the antithesis of what it takes to build a successful startup, and I worry that defining Product Engineer this way just gives them another place to hang their hat and further their own preciousness.
"Engineer" should be reserved for someone who codes or at least administers technical systems hands on. I'm well aware that traditional engineering disciplines look down their nose at programmers for declaring themselves engineers with such informal practices, but nevertheless software engineering is a thing with its own challenges (in many ways more difficult than physical engineering due to the breadth of scale and lack of constraints). Why do we need to water it down when we already have terms like Product Manager or UX Designer?