I can’t fathom this being an issue in 2025 where you can have digitized versions of your kid’s birth certificate on your phone, which indicate that you are their parent.
The kidnapping thing has nothing to do with names, if you only have one parent every country has their rules and you should check them out ahead of time.
We did it with every kid/parent and name combination. The only time we where ever asked to show papers was when the names didn't match. It shouldn't be that way, but IME it's a factor.
As you point out, being the parent doesn't really matter in that case, it needs to be proven that both parent agree on the kid leaving the country.
The kidnapping thing has nothing to do with names, if you only have one parent every country has their rules and you should check them out ahead of time.