At the intersection of personality and mechanics
The Player's Handbook on Intelligence, Wisdom, and Charisma says
You can use your character’s Intelligence, Wisdom, and Charisma scores
to guide you in roleplaying your character. Here is some background (just guidelines) about what these scores can mean. (10)
(Emphasis mine.) Then the Player's Handbook details what possible effects those high and low scores in each mental ability may have on a character's personality. In this DM's case, when players were creating characters, it sounds as if players were supposed to ignore their characters' ability scores because their PCs don't know their ability scores. While the game says that's fine—it says right there in the quotation that you can rather than you must—, totally ignoring ability scores when creating a character is difficult because the character should already know at least some of her strengths and weaknesses. Further, ignoring ability scores when playing a character means the character is, to some degree, actively denying reality.
Most characters should have some sense of their relative scores
This DM thinks he understands what that DM is aiming for when that DM asks that players avoid making role-playing decisions for their PCs based on the player's knowledge of the PC's ability scores: that DM wants the role-playing experience to be more organic, focusing more on narrative than numbers. For example, this DM suspects that DM wants PCs to act and react according to the PCs' personalities and positions rather than their character sheets. And, to a degree, that's legit.
But a character with, for example, a low Charisma score really should know that some folks that possess greater force of personality—y'know, actual and for-real-in-the-campaign-world charisma—are just better at some things than she is. Although she may or may not be aware of her actual Charisma score, all but the most oblivious character would probably know the following:
…And that's only skills and only some skills that have as their key ability Charisma! Of course, there's certainly a role-playing opportunity to be had that involves playing a self-deluded character that can't put two and two together and fails to take into account a lifetime of such realities, but that's an extremely specific role-playing opportunity. Mandating such ignorance among the entire campaign's populace generally seems—to this DM, anyway—a very difficult challenge.
However, this doesn't mean this DM is wholly unsympathetic to your DM's mandate! Verisimilitude in a fantasy campaign is difficult to achieve, and a DM that says, "Play the character not the sheet!" is making a relatively reasonable demand if that demand is then unaffected by mechanics.
I mean, for example, it may rub the DM wrong to have players determine whose PC has the highest Charisma then have their PCs pick that PC to address the king because that PC has the highest Charisma! The DM may feel like the story would be better—or, at least, that it would possess greater verisimilitude—were the PCs to decide that the Charisma 6 dwarf fighter who's a knight of the realm from a family with deep roots in the campaign world should address the king rather than the Charisma 18 sorcerer who's the vagabond offspring of a charming beggar and wayward, outcast half-eladrin.
So, under such a character-not-sheet! mandate, in the previous example the players may have their PCs pick to have the dwarf knight address the king rather than the aasimar sorcerer because, frankly, that makes sense to the PCs. Then if the king says, "Ah, a noble knight of the realm and a member of House Beardaxe! Your renown and lineage precede you! Request granted!" then the players'll have their PCs do that more often. And if the king says, "Wow, aren't you a terrible knight! Gruff, unpleasant, and possessed of terrible personal hygiene! Request denied!" then the players won't. But that kind of trust is usually earned rather than mandated.
Note: Ability score loss alone convinces this DM that a character-not-sheet! mandate would likely very quickly cause controversy among his players. Mandating players make decisions for their PCs based on the belief that PCs have no concept of their own ability scores means overruling players' decisions about their PCs' actions in favor of what the DM thinks is more believable, plausible, or true, and that's something this DM tries hard to avoid. Seriously, I already get to play the rest of the universe so I tend to let players play their PCs however they want.