The community and Wizards of the Coast have different use-cases and goals for their abbreviations. If either blindly followed the other, the abbreviations wouldn’t be doing their respective jobs.
To begin, this split is at least 20 years old. And as that list demonstrates, it doesn’t end with the Player’s Handbook—several books have such dichotomies. In the “v.3.5 revised edition” of D&D, we see “SC” used by the community exclusively for Spell Compendium, while officially that’s the abbreviation for The Sunless Citadel. Officially, Spell Compendium is “SpC” but that is rarely used by the community, and the community’s abbreviation for The Sunless Citadel is simply that the community doesn’t abbreviate The Sunless Citadel, it’s simply not referred to often enough for it. So that’s a much more serious difference than the Player’s Handbook has.
Why? Because the community and Wizards of the Coast have different purposes for their abbreviations.
Wizards of the Coast uses the abbreviations in printed material—stuff they cannot change later. They also seem to use them as product codes, which are also fixed and inflexible—and require uniqueness absent of context. To meet these needs, there are some rules evident in their abbreviations:
Abbreviations are at least 2 letters, but otherwise they are as short as possible.
Abbreviations are assigned on a “first come, first served” basis.
Every product must have an abbreviation.
Abbreviations absolutely cannot be reused.
That’s why the Player’s Handbook gets “PH” and Planar Handbook gets “PlH”—Player’s Handbook was published first, and “PH” is the obvious minimum two-letter abbreviation for it. It’s also why The Sunless Citadel is abbreviated “SC”—it came out long before Spell Compendium. When Spell Compendium came out, Wizards of the Coast couldn’t use SC for it—it had already used that for The Sunless Citadel. Spell Compendium had to be SpC.
The community does not have any of these requirements. Abbreviations are used ephemerally, for one discussion or post, and can be different from context to context. If a later publication is a better candidate for an abbreviation, it can take that one from then on, and even if some people keep using it for the old thing, context can usually handle things. Plenty of publications don’t need an abbreviation at all. So those free the community up in ways Wizards of the Coast isn’t. On top of that, the community has one iron-clad priority that is decidedly secondary for Wizards of the Coast: abbreviations must be clear without having to look them up. They don’t save time or effort otherwise.
As a result, it’s entirely sensible and expected that the community abbreviations won’t match the official ones. Why does “PHB” feel more natural to players than “PH”? As Kirt’s fine answer and Quadratic Wizard’s—as well as comments by VLAZ and Nobody the Hobgoblin—indicate, “PHB” is what TSR used for the Player’s Handbook going back to the original in 1978. Changing the abbreviation would have been confusing—and that violates the community’s cardinal rule.