Softmax is an activation function. The purpose is not just to ensure that the values are normalized (or rescaled) to sum = 1, but also allow to be used as input to cross-entropy loss (hence the function needs to be differentiable).
For your case, the inputs can be arbitrary values (not necessarily probability vectors). It is possible that there’s a mix of positive and negative values which still sum = 1 (eg: [0.3, 0.8, -0.2].
Since softmax picks the class with the highest value, with the values being softly rescaled, hence the name ‘soft’-‘max’.