Cecil Adams tackles the question, Why are girls and women not given the title “Junior,” “II,” etc.?
First he says it rarely happens, and then he gives a few examples of when it does. But his examples are mostly not very good ones. Winifred Sackville Stoner, Jr. - yes, that was the name she was known by. Not some of the others.
FDR's wife and daughter, both legally Anna Eleanor Roosevelt, are only distinguished as "Sr." and "2nd" in legal documents. The mother was known as Eleanor, as everyone jolly well knows (and possibly because her mother's name was Anna), and the daughter was known as Anna, so in general usage, even before Anna's marriage, there was no possibility of confusion.
Dorothy Fuldheim's daughter the professor, I don't know how she was referred to before her marriage, but she bylined her Ph.D. thesis Dorothy Fuldheim Urman. Unfortunately, she died a couple years later, so that's her only publication I could find.
Nancy Sinatra is sometimes called "Jr." in news stories, but I don't believe she's ever used the addition on her albums.
Here's a datum to add to the list. I have a friend who is known to her family as June, not because her name is June - it isn't - but because she has the same first name as her mother and is hence a Junior. In bylines and to her friends she uses her legal name, which is Edith, but without a "Jr." appended, not that there'd be any likelihood of confusion anyway.