In large cities especially, libraries will have an "independent" charitable foundation attached to them. It is very prestigious in the local community to be on the board or to donate large sums to these foundations. Thus, large city libraries are typically excellent with plenty of funding and can afford to offer services to the poor and indigent which smaller libraries cannot.
In the US, a lot of libraries are funded by property taxes, and the various laws that allow these taxes to be collected (for library purposes) will state that the library cannot offer services to people outside the geographic boundary for a lower cost than is charged to the people inside the boundary and are paying the taxes. That's why they need "proof" of where you live before giving you a card. But then you also have laws requiring services to be provided to the homeless regardless of proof of residence (how do you prove your residence when you are homeless?). How does the library resolve that legal conflict? It pretty much always comes down to money and local attitudes (see first paragraph).
In the US, a lot of libraries are funded by property taxes, and the various laws that allow these taxes to be collected (for library purposes) will state that the library cannot offer services to people outside the geographic boundary for a lower cost than is charged to the people inside the boundary and are paying the taxes. That's why they need "proof" of where you live before giving you a card. But then you also have laws requiring services to be provided to the homeless regardless of proof of residence (how do you prove your residence when you are homeless?). How does the library resolve that legal conflict? It pretty much always comes down to money and local attitudes (see first paragraph).