Do we need a definition of life to classify objects? If so, is this definition limited to Earth?

If you mean a split into living versus non-living, no we don't. You can come up with whatever scheme you like to classify things. However, we tend to like classifications that reflect evolutionary history not human perceptions. We would hope that biologists from another planet would come up with the same classifications we do.

Thus the definitions are not limited to Earth.

"Hope is a duty from which palaeontologists are exempt."
David Quammen