I understand your concerns and share your professional views. We as nurses are required by law in most states to identify ourselves, with at least a name badge, when we are "on the clock". I work in a physician's office and asssist with endoscopic procedures daily.
If I choose to wear clothing other than a "uniform", I ALWAYS wear my white jacket and name badge. If I am going to be involved endoscopy that day, I wear my protective gown , shoe covers, mask, gloves, etc. as well.
I had a similar experience to yours when I was admitting my brother-in-law to a physical rehab. facility. It was casual Friday at this facility too, and this woman came in to do his intake and physicl exam. At no point did she identify who she was or what her qualifications were. She was wearing jeans and a plaid shirt with boots!
I was very weary of the facility from this point.