Sometimes people are just there to help others; but too often they're chancers who, while helping others, will also help themselves.
It's not easy to tell the difference - if it were, scammers wouldn't get very far! But what DOES make sense is to be cautious, to be wary, and to try and get the OP's mum to establish boundaries, whereby "Anne" doesn't have direct access to bank account details, credit/debit cards or anything else that will allow her to drain the OP's mum's account.
If the OP says to her mum "Look, I can do all your insurance and bill paying online for you, we'll set it up together and then you won't need to worry about it, or worry "Anne" about it either" then that might be the safest way to go about it. Removes the opportunity for "Anne" to have ongoing access to the accounts and personal details. And of course, "Anne" must never have passwords to the online stuff.
It is also, at her age, a very good idea to get the POA sorted out before it becomes too late. Someone has already said on this thread that their elderly relative was about to be shipped off to a solicitor with her "friends", and that was only stopped in the nick of time - so talk to your mother about POA and help her understand that it's not about taking power away from her, it's about being able to make decisions for her when, and only when, she becomes unable to make those decisions by herself. It can be a hard conversation to have and often involves other health decisions (such as DNR choices) - but it's better to do it now while she's still able than to be stuck when she's not.
My Dad has EPOA for a couple of very old friends, who have no other relatives; and has had it for elderly relatives as well. He is 84 himself!