This may just be a coincidence and actual be two different people.
Our Anna was born in Stockholm September 14, 1865. Family history says her father was a butcher and had a large family with at least two children being Hurburt and Hildah with unknown birth dates.
The three children settled in California near San Francisco sometime before 1890. So we were told that Anna came from Sweden to America (California) twice before 1890.
There are a couple possible records showing an Anna Anderson travelling to America that I have found reference to, however I no longer have the full Ancestry.com account so I no longer have access to see if Hildah and Huburt were listed along on one of these:
Gothenburg, Sweden, Passenger Lists, 1869-1951 (in Swedish) Record Immigration & Travel
Name:Anna M Andersson Birth:abt 1865 in Stockholm Arrival:New York, New York Departure:1 Nov. 1886
Name:Anna Maria Anderson Birth:abt 1864 in Stockholm Arrival:Hull, England Departure:8 Maj 1891
Name:Anna Anderson Birth:abt 1865 in Stockholm Arrival:Hull, England Departure:24 Febr. 1882
Also I know there was another Anna M Anderson which I keep bumping into who was in Michigan by 1920's which is not our Anna.
Finally there is information that may be connected to our Anna found in:
Sweden, Births from the Swedish Death Index, 1901-2006 (in Swedish) Record