All three are correct, but they mean different things. The third one is much more common than the other two.
The third sentence means "her ability to swim was greater than or equal to his." If he can swim 100 meters freestyle in 59 seconds, then she believes that she can also swim 100 meters in 59 seconds, and can perhaps swim even faster. She believes that she can beat him or at least tie with him in a race.
The other two sentences are different. They both create the category "good swimmers". Some people are good swimmers, and some people are not.
In sentence 1, she ALREADY KNOWS that he is a good swimmer, and she BELIEVES that she is also a good swimmer, but she is not certain.
In sentence 2, she BELIEVES that she and he are both good swimmers, but she is not certain.
So, in sentence 1, she KNOWS that he is a good swimmer, while in sentence 2, she only believes it.
WHY?
X is as [adj] as Y= X's adj is greater than or equal to Y's adj
One of the things that inversion can do is subordinate a clause to the primary verb of the previous clause. In sentence 1, the two clauses are independent: you can say them in either order:
1=He was a good swimmer, and she felt that she was one, too.
However, by reversing the noun and verb in the second clause, you make the second clause subordinate to the first, so it is equivalent to repeating the verb "felt"
2= She felt that she was a good swimmer, and she also felt that he was a good swimmer.