Also being the devil's advocate
, I believe it is not residency, since the policy clearly states it is NOT required, but more being a member of the household.
As a 50 year old visiting your mother in the town you are clearly NOT a member of the household, merely a visitor.
On the other hand, the 18 year old is still basically a member of her parents household. Based on your reasoning, a 14 year old up for the summer at her parents house would not be entitled to a free card. Neither would a resident's child since SHE is not the actual taxpayer.
What urgent principle is the library protecting here? When voting we want to make sure every entitled person can vote, but only once. Some certification is reasonable. But the library should be in the business of dispensing knowledge and culture. What difference does it make if a few household members of taxpayers take out materials?
So if you were living with your mother, even at 50, you WOULD be a member of her household and it would be reasonable to issue you a library card. What better goal is served by refusing you one?
My critique of the librarian, or actually of the apparent library policy, is that I would think librarians would want to get books into as many hands as they possibly could. This interpretation of the policy seems to undermine that idea.