I suspect EllieFAntspoo deliberately feeds the neighbour false information, in case said neighbour turns out to be a spy or a blabbermouth. The best course of action in life is never to tell anyone the truth about anything. The walls have ears.
Yes, of course the more information someone has about you, and the easier it is for them to find it, assuming everything you have said about yourself is true, which often isn't the case, the easier they will find it to pretend to be you. However, to get access to your money, they don't actually need much information at all. Besides which, your most sensitive and accurate information is held by notoriously leaky services, like HMRC, the NHS, the Passport Agency, banks (which have been known to leave client details lying about in rubbish sacks outside their premises), the General Register Office etc, so identity theft really could happen to anyone who is known to live here legally. You only really have to give your information to an organisation once to have increased your risk that this information will find its way into the wrong hands.
The ideal identity theft, of course, is of someone who is already dead, as then you don't have to risk bumping into them. Whoever it happens to, it's an awful thing to happen and there is little point being smug and pretending they deserved it, because actually, it can happen even to those who think they are being careful. So yes, do think about what you are telling people online, but don't think that by being careful you are therefore safe, because you aren't, and if you worry about that fact too much, you will spend your life in a state of paranoid anxiety.