Interesting question!
I do think that an entirely new taxonomic classification, separate from the one classifying earthly life would need to be created. Perhaps, in order to be able to re-use names that happen to be descriptive of the newly found life as well as earthly life, a whole new top (or bottom, however you want to view it) level classification would have to be added that deliniated Eartly life and the newly discovered planet X life.
So that, say "Terran" would be added to the full scientific name of eartly species, and similarly some other title would be added to life found, say, under the Ice on the moon Europa.
I don't think the extraterrestrial life could reasonably be classified within the same system as earthly life, as common descent is a crucial criteria, I think. Though things would get REALLY interesting if primitive life was found elsewhere in our solar system that had the same basic biochemistry, and used DNA with a genetic code not significantly different from that of earthly life... in that case I think we would be left with the question of whether the particular genetic code, and biochemistry that happens in Earthly life is the ONLY kind of chemistry that life could be based on, or whether perhaps the panspermia hypothesis, that primitive bacteria-like life rained down on the early Earth from space would have to be accepted. In the latter case there WOULD be common descent between earthly life and extraterrestrial life, and the tree of life would have to be split right at the root, to give a new brach for the extraterrestrial life.
Whatever the case, the discovery of extraterrestrial life, even the simplest kind, would be a massive boon to our understanding of life.
If we found extraterrestrial life what would we call it and more specifically how would we classify it.
This question may sound trite but it is not meant to be. For example what is an animal, a plant or a fungi. We probably all have a picture in our mind of examples of these organisms but what most people don't really think about is that an animal is any representative of kingdom animalia, a plant is a rep. of k. plantae, and a fungi is a rep. of kingdom fungi. To be a representative of a taxonomic group is to share an evolutionary heritage with other members of that group.
Suppose for a moment that in the future some team of exobiologists sent to explore another planet discover life that looks vaguely similar to that which we know here from earth. If they found a scaly aquatic cold-blooded pisciform shaped organism with an internal sceletal structure the media would doubtless call it a fish, but could science? All fish share a (hypothetical) common ancestor but it stretches reason to suppose that our newly discovered quasi-fish could have the same ancestry.
With complex life forms it is perhaps very unlikely that a planet with the exact initial starting conditions as earth would end up with complex life forms which were at all similar to ours, but it seems to me that the simpler the life form then the more likelyhood there might be that evolution would arrive at a similar "body plan". If we found bacteria like organisms on another planet, and it could be reasonably proven that they were not contaminants and that they did not likely arrive via panspermia (or vice versa) then could we really call them bacteria.
Note for purists: I don't think that the word bacteria has much taxonomic significance these days since it is rather vague. Just use "archaea" or "eubacteria" instead of the word "bacteria".
Well, what do you think?