Only by some, apparently.
Yes, by her aunt Olga and her uncle Andrew. Her aunt Irene was doubtful, and no other close relatives saw her. Xenia Leeds recognized her, but had not seen her since they were children and based her recognition on AA's memory and general behaviour. But the Botkin children who were among the last to see her, recognized her immediately.
If Princess Mary had disappeared late in World War I at a time when the royal family was alleged to have been executed and then suddenly showed up four or five years later, or if Princess Margaret had done so after World War II, I can't imagine that there'd have been any doubt in anyone's mind. The two princesses looked so similar at age 17 or 18 and in their mid-20s; it's not as though Anastasia disappeared at the age of 10 and reappeared at the age of 35, when there would have been some significant changes.
You forget that Anastasia was a rather plump teenager who later showed up as thin as a rail with all her upper front teeth missing and her jaws broken. At the time Olga met her, she was deadly ill and on morphine several times a day. I have myself met a cancer patient and not recognized him even though less than a year had passed since the last time I saw him. Also, remember Gilliard who met Volkov again after a few months, and he did not recognize him.
Yet for every family member who claims that Anna Anderson was Anastasia, there seems to be one who's sure she isn't. I just can't imagine that much doubt surrounding Princess Margaret in 1948 or Princess Mary in 1922.
As I have already pointed out: Not many of her close family met her. Her grandmother would not receive her, Uncle Ernie refused to travel to Berlin to see her, aunt Xenia even cabled Olga and told her not to acknowledge the unknown woman in Berlin.
As for Peter Kurth being an expert on Anastasia and shrugging off everyone else's expertise - he's the guy who pooh-poohed the DNA evidence from Prince Philip on account of a lack of witnesses when the hair sample was collected, even though the sample in question was a blood sample, not a hair sample. In contrast, the scientists who did the DNA tests actually ARE experts.
This you have to discuss with Peter Kurth.