I see what you mean, but I don't think I agree. I don't think that two versions of the "same" rhyme or song constitute a psychological gestalt so that all meanings of either are meanings of both. I feel that the one with the tiger refers to a large Eurasian cat, especially if the speaker and hearer are unaware of other versions that refer to a person of African descent.
Incidentally, folklorists (of which I am one) have made the same argument you put forth, calling it "the symbolic equivalence of allomotifs." It's very much a respected position on these matters and I'm not belittling it. I just don't happen to agree in this case.