Georgiana's birthmark in Hawthorne's short story "The Birthmark" is shaped like a small hand. Those who view the birthmark are of two minds; some think it is a shame that this birthmark mars an otherwise absolutely perfect face while others think it looks like a little fairy hand. (Interestingly, most men think the birthmark enhances Georgiana's beauty, while jealous women say it is unattractive.) Those who find the birthmark charming imagine that a fairy touched Georgiana at birth, giving her not only her transcendant beauty but her loving disposition as well. Georgiana herself liked the birthmark before her marriage to Aylmer.
Unfortunately, Aylmer is part of the former group and wants to do anything to rid Georgiana of her birthmark so that she will be perfect. He dreams that he tries to surgically remove the birthmark, only to have it enlarge until it is part of her heart (possibly foreshadowing the story's end). Eventually, with the help of his assistant Aminadab, Aylmer concocts a potion that removes the birthmark, but it takes Georgiana's life, too. The message here is that perfection is not consistent with human life. Achieving perfection, in this case, means sacrificing the Georgiana's life.
Some critics, such as Susan Howe, find deeper symbolism in the crimson hand, likening its symbolism to the whale of Moby Dick and the crimson A in The Scarlet Letter. Howe states (in the e-notes essay cited below):
This "singular mark" wears "a tint of deeper crimson," bears "not a little similarity to the human hand," and signifies endlessly (p. 765). In the course of only a few pages, the hand is referred to as "the Bloody Hand," "the Crimson Hand," and "the odious Hand" and is likened to a "fairy sign-manual" (pp. 765, 766, 767). The hand, in other words, is a text not unlike the scarlet letter or the white whale, about which Ahab says, "he tasks me; he heaps me" (p. 164). Like Moby-Dick, the birthmark, both the physical mark and the story itself, begs to be read.
Howe goes on to liken the crimson birthmark to female sexuality, noting that Aylmer's distaste for the birthmark doesn't begin to grow until after the couple has married and has consumated their marriage. Thus, the color of the birthmark is symbolic as well as its shape.
Given the attention given this symbol in literary criticism, it is probably fair to say that a fish or heart birthmark just wouldn't be the same.