Not so much with how D&D has typically presented the races. Dwarves are a different "race," but as of 5e, the only thing that might be called definitively "biological" about being a dwarf is that your CON might be 1 point higher than a human if you both max out CON. But why couldn't a "hardy group of humans" get that max?
I'm assuming that whenever there is a bonus to an ability score, it is because the -physiological/intellectual/magical/mystical/etc- characteristics of the race differ enough that it goes beyond what a "hardy group of humans" could get by normal means (there's always magical/mystical/etc means to attain crazy stuff, of course). There are limits that humans cannot go beyond under normal circumstances; no matter how much we train, our bodies are simply not mechanically capable of ever matching the strength of a 1,000-pound gorilla without some kind of augmentation. There might be a way to train in such a way to overcome that strength difference through other means (some ape-reducing martial art, I don't know), but merely by their passive nature, it would be very reasonable for a sapient gorilla PC to have a higher potential STR than a regular human PC.
How granular do you want your fantasy? Do your fireballs displace air? Does your STR score model the distinction between explosive force and long-term structure and hand-eye coordination?
In practice, a lot of these differences don't matter at the level of mechanics in our game of dragon men and magical elves. Certainly I'm not quibbling over the explosive force differential between two roughly-similarly-shaped apes when there's undead walkin' around who can huck a spear without any muscles.
It doesn't have to go to such extremes; I'm speaking about characteristics that are salient enough as to warrant differences that are already expressed by the rules (or, more precisely to avoid going off a tangent, I'm trying to argue in favour of the existence of said mechanical differences). In the sapient gorilla vs human case it can be a noticeable difference in musculature that justifies a STR bonus/higher threshold; in the case of dwarfs vs humans it could be a very different immune system or preternatural resistance that explains their higher CON.
Difference settings might have entire different reasons, or have no mechanical differences between the classical races if it fits with its internal logic. The point is that two different kinds of races, even two different kinds of similar humanoids like dwarfs and humans, can be substantially divergent as to warrant mechanical differences within the D&D rule system, enough that just training and trying hard enough won't make up for them.
There's nothin' wrong with "Strong" as a character archetype, or even "Strong" being something folks usually pin on a given fantasy race, but whether or not your character is "Strong" should be a player choice, not something made by the fantasy group to which they belong. Maxing out your Strength should be the same for every character, so that when the Gorilla-Person meets the Mighty Barbarian they can get in an arm-wrestling contest and you won't know who's gonna win!
I agree with the principle, but it should depend on the circumstances of the game. Being able to choose whether to play a "Strong" character should certainly be within the prerogative of the player (to which degree depends on the type of game, of course), but that shouldn't be at the expense of the setting's internal consistency. And in D&D at least, where a degree of simulationism is always at hand, the player should in some capacity have to conform to the limits imposed by the nature of the races in a particular setting (which brings me to the arguments I make in the previous part of the post). That doesn't mean sticking to the same trope (subverting them is half the fun, after all), but it does mean that if there is a race of Half-Giants with a +4 STR, with a reasonable explanation as to why such difference exists, a player shouldn't expect to achieve the same level of brute force with a Gnome without some manner of additional explanation (such as augmentations or preternatural powers. An explanation of "he's really, really heroic" would work too, assuming it fits the internal consistency of the game, like in, say, the Adventures of the Baron of Münchhausen).