No. Classification result using the same algorithm are not different. If it comes different implementation is not correct.

Initial values (usually initial cluster centers) are important. Usually these values are chosen randomly at the beginning of the clustering which cause same algorithm acts differently in different softwares, or even different runs.

It depends on which version of KMeans you are using. If your KMeans uses random centroids naturally different results will be obtained in the same algorithm. Even for other versions of KMeans that are not stochastic they can differentiate in the way they choose centroids, the quantity of them or in the way they group the points.