The island of New Guinea is split by an almost-straight line into Indonesia (Asia) and Papua New Guinea (Oceania/Pacific). How did such a neat border arise? How does an island come to be split not just between countries but between continents?

Someone conquered part of it and got bored.

No, really.

Okay, not really. But actually kinda.

The current division is the result of colonial endeavours, surprise surprise. The Eastern half had been changing hands since the 1880s. The northern half of PNG was one of the earliest colonies of the German Empire starting in 1884, but only lasted 30 years before being taken by Australia. Australian interest predates WWI but not to such an extent that any notable effort was made to take the eastern half of the island before then.

At that time, the south of what is now PNG was in British hands (eventually under the administration of Queensland) while the Dutch held the West going back to 1828. By WWI Australia claimed all of the eastern half and by 1920 the Dutch were formally administering the western half. They maintained control of it until 1962.

But actually it kinda is because "someone conquered part and got bored". The earlier claim was that of the Netherlands in 1828 and they only claimed the western half.

A big part of the limit of their claim has to do with the history of the VOC in the area two centuries earlier. Ternate had been a Dutch ally and as of the 1660s they also exerted pressure on Tidore, though this wasn't as effective until later. Anyway it was ultimately the Tidore connection to Paupa through the legacy of Nuku Muhammad Amiruddin that was used to justify the Netherland's claims on the island with the "sphere of influence" argument, with permission from the Sultanate itself. Tidore, after making their own claims on the island, granted the Dutch permission to set up shop in 1872.

The remainder of the island east of the 141st meridian was then up for grabs, as it were, which is where Germany and Britain (later Australia) come in.

So, as you see, they unfortunately kinda claimed half and then got bored.

note: when I say PNG I really just mean the eastern half of the island, roughly what is today PNG.

Disclosure: I'm not a PNG historian at all, but I am a linguist and PNG figures heavily in my department, and Ternate and Tidore are also linguistically important places, so I've been exposed to a reasonable amount of information about these places. I more than welcome a more in-depth response if there's someone who can give one.

edit: typos