The problem with "ethical" or "unethical" is that circumstances can have an influence. For example, let's say a regular, reliable client suddenly doesn't show up for his usual therapy session. The second time it happens the therapist is concerned. So she looks into the client's records and sees there is no emergency contact. Because she is concerned she googles his name, hoping to find his phone number or maybe work place so she can inquire as to his well-being.
I would not call that unethical.
But my very own opinion? I've learned that in this year of 2022 and henceforth, it is smart for anyone doing any kind of professional work with someone - and even someone considering a prospective friendship or date - to google the person.