They do not have to play 'dumb as a post' or 'trips over their tongue when they talk' characters, but I will be damned if they don't roleplay appropriate limitations at all. That is very poor roleplay, and poor gamesmanship, and I don't let it slide at my table as GM.
There was a lot of ink spilled long ago about character's having dump stats (we're talking 3-6 range low) that didn't seem to affect them in one bit as far as RP was concerned. In AD&D, this was especially common as the ability modifiers for mental scores tended to only affect niche areas (languages, cham saves, henchmen) unless you were a spellcaster. Stories were told of dwarven fighters or barbarians with single-digit ability scores being as clever, learned, or smooth as the player was capable of role-playing (a form of metagaming in a way) which created the backlash of "roleplaying your score" to make characters stupid, gullible, or ugly/unsociable. It stuck for a long time and now it appears the trend is starting to reverse.
here’s this weird double standard where physical stats speak for themselves and mental/social ones have to be “roleplayed” according to whatever standard the DM decides.
Some role-players actually role-play low physical stats too. A long strength PC might (voluntarily) have a hard time lifting or moving a heavy object (even if the DM didn't call for a strength check) or intentionally climb up something slowly. Low dex PCs might be clumsy, bumping into people or objects. Low Con PCs might be sickly, with permanent coughs, easily catching cold from drafts, or nauseated easily by food or drink. But just like the above, plenty of players don't look at the ability score as anything but a mechanical adjustment. The low strength wizard will attempt to open stuck doors and rely on that "nat 20" to bypass the -1 to the roll. Ditto for low dex or con.
Its already common knowledge that the 3-18 score range is an artifact that serves little purpose anymore, the ability modifier (-5 to +5) is what is important. Plenty of d20 Variants removed it to no great loss. You could easily replicate granting the same modifier range without needing Str/Dex/Con/Int/Wis/Cha and fix a whole lot of problems with the game in one fell swoop. The biggest would be trying to numerically quantify how strong/smart/agile someone is and try to explain what the difference between a 12 and a 13 Dexterity is supposed to represent.
All I get from this is that, we should only play our self in game.
Not quite. But I'm getting the vibe in certain gaming circles that D&D is still too focused on numbers and making them mean something. For example, much of the hoopla about racial ASI was the notion that ideas like "strong" or "small" or "graceful" should have corresponding mechanical, numeric representation in the game. A strong race should have a bonus to Strength, for example. However, if you remove the concept of Strength as a mechanic, then "strong" race no longer has a mechanical expression, it's strictly a role-playing tool. Or if Strength exists, maybe it shouldn't affect other systems mechanically (such as bonus to hit and damage) since a.) no one agrees if using a longsword relies of muscle or agility anyway and b.) it would break the notion that warriors are strong and strong races are the best warriors.
Once you remove specific numeric expressions of a character's abilities, you open a huge amount of potential to design and play characters however you like.