Sounds silly, you are right. A neighbour was interviewed and said that she (the deceased) had lived there for a month and that nobody had spoken to her or knew who she was.
I don't know about Mexico, but here (Denmark) you are not legally obliged to give your name when you rent something, even if it would be highly unusual to get a flat without giving a name. But you could call yourself Jane Doe - few would ask for legitimation, if you look credible and pay in advance. You are obliged to report your new adress to a general public register within five days - but it will take some time before they find out that you didn't. And I don't think they have that kind of register in Mexico City with 20+ millions of inhabitants.
So without knowing it, I would guess that it is possible to live quite anonymously in Mexico City (and elsewhere in the country).
There is of course also the possibility that they just brought the news before the police had contacted the houseowner and got the name on the contract (if there is one). Then the identity might show up in the blogs, but don't be sure: The jumper is a better story than her name.
And kudos to you for seeing the obvious problem

- to be honest, I didn't think that far

Holy crap wouldn't that just be breeding grounds for gangs, terrorists and other things? In NZ most legitimate agencies require references and ID when you even apply to rent a place, private landlords may be more relaxed about it but no one wants to risk renting their property to a stranger with no ID.
Shows how different other people in other parts of the world live!