I think you've put your finger on another piece of poor explanation twice over. The first and obvious part is that if you are using a battlemap the characters (PC and NPC alike) are neither more nor less fixed in where they are than in any other version - just as characters in films with continuity are fixed in their physical locations whether the background is real or a green screen. But on a green screen the characters aren't going to interact with elements of the set, and we know they aren't because they aren't really there.
But more important is that you can reskin entire characters and you can do this because outside the basic class features no power is mandatory. Which means you can pick and choose a lot more things that are near enough that they fit.
To take an extreme hypothetical example let's assume that I want to play a goblin on a pogo stick in a comedy game. I want to be nippy and nimble, and bounce around, and I don't want to be an extreme high hit point tank or a spellcaster. There are two obvious choices for nippy classes which aren't tanks or spellcasters; rogue and monk. I don't think that an absurd sneak attack every time fits - so rogue doesn't fit that well (I might dip a level or two if I were to try this in 3.5 or 5e but 3d6 sneak attacks are right out).
Playing my goblin on a pogo stick as a monk is just about OK for the first four levels; the ki pool is a bit odd for this chaotic a character - but then I hit fifth level with Stunning Fist and sixth which gives me unarmed magic weapons. I can just about justify the stunning attack as hopping on someone's head - but stunning fist is so good you normally spam it, which means that I either become a one note joke . Even the normal ki pool is a little odd. It just feels wrong.
Meanwhile in 4e the Monk has no mandatory abilities. We instead have a number of different powers to pick from different visions of the monk. Let's for the sake of argument call them "Focused warrior", "Four Elements", "Wire-Fu", and "Drunken Master" (the historical fighting style being based round pretending to be drunk although you can take the more cinematic approach). All the monk powers come from one or other of these approaches - and this is where 4e becomes a lot easier to create reskinned characters than any other edition; the 3.5 and 5e monks are based almost entirely round the focused warrior archetype. 5e can also have an additional layer (such as four elements or ninja) or double down on the focused warrior. But 4e we can take our powers from any of our archetypes, which means our goblin on a pogo stick gets to be an almost pure mix of wire-fu and drunken master with none of the focused warrior to be seen (it's a goblin on a pogo stick in a battle - do you really expect focus).
To illustrate I look at the class feature, and the first Four Elements choice, Desert Wind, doing fire damage is completely wrong - but Centered Breath that when you hit someone you can slide an enemy around you (whether or not it's the one you hit) is perfect for the chaotic goblin on a pogo stick. Likewise when it comes to picking my at wills, the four elements choice blistering flourish (doing fire damage and with a move action that burns enemies that hit you with opportunity attacks) would not fit. But one of my at wills being Dragon's Tail (with an attack that knocks enemies down and a move action that lets you swap spaces with an enemy or a prone ally) is perfect for the effects of a pogo stick jumping on the enemy. And Fallen Needle (discombobulating enemies with the attack for a -2 to hit you if you hit them, and a minor action move or shift) is superb to represent the pogo stick bounding off in random directions. So by the time we've picked our at wills we know our goblin is fast and slippery - and likes bouncing on peoples' heads to knock them down. This is what they do when they aren't sure what else to do.
And then we pick encounter and daily powers again to match our ridiculous pogo-sticking that leaves us bouncing around to ludicrous places and leaving enemies confused or flattened. Because all our powers are strictly powers we have picked to fit the character concept and there is nothing we had to take the whole thing fits much better. It's not that we could reskin more easily, but that we could easily duck anything that didn't fit the concept in a way you simply can't in other editions.
And this is a criticism I can sympathise with. I never found it that bad in part because I've got a head for things like that - but I found the issue vanished when I started building concepts that fit the genre I was going for and then picking powers based on that concept. The powers were just how I represented what my character did and the picks flowed from the concept. Which meant that the choices flowed from what I visualised my character doing in related situations.
And what balance means to 4e players is that a concept as ridiculous as a Blood Bowl inspired goblin on a pogo stick will not only be buildable and effective because of 4e's unmatched flexibility in character design but I know I can do it without letting everyone else in the group down by playing a dud character who isn't able to hold their own (but is highly distinctive in what they do). So the only thing I need to worry about is whether it's a bad fit for the game and the table.
