That specific example is a bad one as you say. A better is perhaps student loans. I recommend checking out BBCs More or less, they has covered the issues with Oxfams claims several times.
Even in the case of student loans, that's still not an incorrect way to count net worth. The only problem with it is that we don't have a good way to measure the value of human capital.
I believe the point stands. If a penniless, homeless child has a worth of 0, and I have a house valued at 80,000 with a 90,000 mortgage, am I really worse off than the child without a home? Assuming I have a job and can make payments.
If you assume you can make payments, then you are equivalently assuming your job has present asset value (or that you have human capital). So yes, a full accounting will show you are not actually underwater.
If you are delusional and have no job security, then yes, you're worse off than the child.