I suppose one simple answer could be that, as the head of the Oxyrhynchus Papyri project, at least as far as literary papyri were concerned, he was presumably the most obvious person to ask. But we don’t know whether other papyrologists were approached before him, so this is just speculation on my part.
At the moment it does not seem we can know for sure whether the owner had already identified the author, but this is not that unlikely: if he had any notion of Greek literature, however vague, he should have recognised even simply from the layout of the papyrus that it was a series of Sapphic stanzas, which gave at least a hint towards authorship.
As far as there being other fragments, I take Dr Obbink’s draft to imply that other fragments of the same papyrus (those classed as “P.GC. inv. 105″) have been identified and will be published at the same time, but apparently they do not belong to the same collector; the “GC” bit in the inventory number has been taken by some to refer to the Green Collection, on whose editorial board – or something of that kind – Dr Obbink reportedly sits.
Needless to say that I look forward to seeing answers to all the questions that have been (rightly) posed in this discussion.