I’m assuming the poster meant refinance with the same terms. So the borrower can get another 40 years to pay back the remaining balance anytime they like.

You're right. I believe the technical term is "recast" where you take all of the outstanding balance and spread it over the course of "the next 40 years" with the same interest rate.

Another fact I omitted is that the interest rate is 2%, but the minimum payment is lower. So the borrower is accumulating owed balance because they are paying less than interesting accruing.