She may have just been genuinely concerned that you would get good care.
It's pretty common for professionals to know each other, and asking for the other's name is Ok. But only to ask once, and if you don't say who, then they should drop it. If she kept asking, then that's going too far.
There's no problem if she respects your dropping the matter and leaves you alone. If she contacts you once more after some time, as a follow up to see how it's going, that's OK and pretty normal. Again as long as she then drops it and leaves you alone.
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