As I see it, you're asking two questions:
Do fictional stories need to have relatable characters in order to be good?
And the answer would be a resounding
But regarding the actual title, Do readers need to identify with fictional characters?
Being able to relate to a character has little to do physical characteristics. Sure, a person who's struggling with disability will be able to relate more with a character who's doing the same (if the character is portrayed well, that is), but that's just a surface level.
Characters are relatable when they show humanity (often even when they're not technically human in their respective universes). With humanity here I mean the "ability to feel human emotion, or perceive the world as an human would do". Despite the fact that humans come in various shapes, gender, sizes, etnicity and culture, the basics of the human experience is worldwide. A person crying in pain will be relatable to each other people who suffered pain before, regardless of the source.
What happens with movies and other visual media is:
- The matter of representation,
- Shortcuts, aka "cues" (there's a more specific term here, but I can't seem to remember it now).
I won't delve into the matter of representation of minorities or different ethnicities since it is indeed a complex issue, and a bit oustide the scope of this answer. I will deal with the shortcuts - also known as "when the representation is done cheap".
In other words, it's when the authors want some share of the audience to relate with a character, but they don't have the time or the will to deal extensively with that character backstory, motivation or psychology. So, in order to make it more interesting to at least a share of the general public, they throw in a "key characteristic" as a token gesture.
In a more general sense, a cue is when you throw hints at the audience expecting them to fill the gaps. Cues can be useful in some situation, but they shouldn't be shortcuts to make your character more relatable.
As you mentioned:
Many movies are starting to introduce black and female characters and even chubby characters, even in situations were it doesn't make sense, because of the argument that the viewer needs to relate to the fictional/historical characters.
I'm recalling Troy: fall of a city link, were suddendly the (supposedly greek) Achilles is portrayed by a black actor (being black of course doesn't influence the skill of the actor, but it doesn't sit well with the historical setting).
The point is that this kind of shortcuts can work - I'd be lying if I told you they don't - but while it's true that you could make your work more "appealing" this way, seeming more inclusive and having a more diverse cast, you also risk alienating some of your audience.
Some of your audience, probably of the same category you're trying to appeal to, will recognize that you're using cheap cues and will feel annoyed. In the end, throwing in, for example, a disabled character without dealing with the complex issues of disability is downright disrespectful.
In the same way, any "characteristic" of the characters you're dealing with should be addressed if it's of some relevance.
Be true to your setting and to the story you want to tell. Relatable character are good characters, and vice-versa; throwing cues won't help making them better.