Can a Power of Attorney (non-family) put herself as the beneficiary?

The POA cannot act in a self-serving manner, they cannot act in their own self-interest. So the answer here is probably NO. NO the POA cannot make themselves a beneficiary. That needs to be done by the principle, without any influence from the POA, if they are still competent.

I was able to open accounts for my brother while he was completely competent, in his name, with his ex partner as POD beneficiary. I was able to renew and roll over accounts in CD with myself as already being POD beneficiary and remaining so upon rollover; I was able to put money into his Trust (I was at that time both beneficiary of the Trust and Trustee of the Trust.) I would not have tried to put myself on an account were he not competent to dictate I do so, as a POA is a Fiduciary, and that person is sworn under law to act on for a person or in that person's best interest. I would have placed them in his name only. As I was his only living relative, only heir, Trustee of his Trust, beneficiary of his Trust, and etc. I would have inherited in any case.
I feel you should not put yourself on any accounts other than as POA without this person dictating you do so and being present when account is opened at his or her bank. You could be liable under the law.
See an Elder Law Attorney or the person who wrote the POA for the elder you are serving with this question. The Elder's estate will pay for this consult as you are acting for the elder, to find out how you must best proceed in his or her interest.

Beneficiary of what?

Generally the answer is no. Any beneficiary designations have to be made by the elder prior to the POA being enacted. Here in Canada a POA cannot change beneficiary designations on bank or investment accounts, or will Wills.

I am not a lawyer but to me this sounds suspect.
If this is happening and it is a family member you might want to consult a lawyer. I would also want to review any and all paperwork this POA has been involved with.