If you look across MGS1, 2, and 3, there's a hint that John is as much a pseudonym as Jack is.

In MGS1, you can't control Snake after he says his real name.

In MGS2, you can't control Raiden after he says the name you gave him isn't his. (cont.) https://twitter.com/HEITAIs/status/1390652851731324929

The series establishes that the separation of player and character hinges on a name. Snake reclaims his own name and Raiden promises to find a name of his own, and we were never meant to control either of them again (except in the tanker for the sake of the fakeout). (cont.)

In MGS3, Naked Snake tells Ocelot his name is John. A few seconds after that, you're in control of him again. (cont.)

"John" lacks the liberating potential of "David" or of Raiden discarding the player's name.

Just like "Solid Snake" or "Naked Snake" or "Raiden" or "Jack," it's a name that allows control. In that sense, it's akin to a codename and not a real name.