I don't really see it as a moral issue, but it seems like bad therapy for a therapist to Google a client. The therapist is supposed to be working based on what the client presents with in therapy, not based on information gleaned from outside sources not introduced by the client. The only exceptions to this would be court mandated clients and minors, but even then it is acknowledged that having that sort exception is not ideal in therapy and has a negative effect.

It might feel weird to have a therapist learn things about you without your consent since many times, you provide official consent before your therapist will talk to someone about you. While your therapist doesn't technically have to get your consent to be on the receiving end of information (so long as they don't confirm you are a client in any way or say anything about you), I can't really think of any way therapists can initiate the receipt of information about clients other than the examples I mentioned that doesn't require client consent. Googling you sort of bypasses the issue since your therapist didn't have to identify you as her client - but I don't think what she did followed the spirit or intention of the rules regarding client consent for the release/exchange/receipt of information.

I think it's weird she Googled where you're from instead of just asking you - especially since she apparently had no problem telling you she was curious about it. I think being interested is good, but not to the point that she can't wait to ask you something until she sees you. Just seems odd.