Linear decision boundaries need not be unique, and in fact, there are infinitely many linear decision boundaries that could classify a linearly separable dataset.

Consider the data and classifiers below. We note that the blue and green classifier both have 0 training error, though the blue classifier probably does not generalize to test data very well.

The assumption and motivation for using SVMs is that by finding the separating hyperplane that both minimizes training error, and maximizes the distance from the plane to the points, we will find a more generalizable model.