"Manhattan" project was the code name for the original XBox.
At the time Japan dominated video consoles with the last western attempt being the disaster which was the 3DO. Thus Microsoft was going "nuke the japanese industry" and gain a foothold.
For further irony the Xbox was in effect was a spiritual successor to Sega's Dreamcast. That Microsoft went with such a distasteful name despite partnering with Sega highlights their cultural issues. Issues which have continued to prevent Xbox from building relationships with Japanese studios.
"Manhattan" was about DirectX, not the XBox. (That's why DirectX had the weird weird radioactive logos, even before the XBox was an idea.)
It wasn't about consoles but gaming in general. Programming Games for DOS required a lot of assembly knowledge (at least, if you wanted something fluid) and deviation from industry norms only increased development cost (thinking things like High res support necessitating VESA code, to say nothing of sound support in DOS.)
Devs hated dealing with all this extra work, extra compatibility, extra libraries (old gamers may recall MILES or DMX setup screens in some titles.) The libraries for Windows 3.1 were... not at all up to the task.
Thus, DirectX was created. Not so much to create a console competitor, as to show PCs as superior to consoles overall.
Which, when one thinks about it, may explain why Eric/etc were not big fans of the XBox; in some ways it conceded defeat on a level.
No it was not a coincidence as testified by Eric in his interview with the University of Washington business school.
He actually comes across a mix of interesting and unpleasant in this interview. It takes me back to the nineties when he and St. John were routinely pissing off people including myself.
According to Eric when he told me about this, it was HIS idea to name it that. When he said this, it made sense because it followed his sense of humor (this was literally in September he told me this story).
Remember that there's HN'ers who never worked at Microsoft :-)