DND_Reborn
The High Aldwin
For me sameyness if having too many PCs being able to do the same thing or have the same feature options. I understand of course some degree of overlap is needed, but too much overlap is when the problems appear IMO. Also, as I stated in the OP, this is primarily about the sameyness between classes, not within a class (where I would expect it).I guess the first question I would have is what qualifies sameness? Is it a diversity of mechanics, a sort of identity tied to each class or a distinctive feel in play for each class? Or is a combination of these things, or, perhaps, something I am completely off the mark of?
An example of acceptable sameyness is Fighting Styles. Fighters get them all. Paladins and Rangers have a reduced subset. Barbarians, Monks, and Rogues have none. This means if a PC has a fighting style, it might be a Fighter, Paladin, or Ranger, but I know it won't be a Barbarian, Monk, or Rogue.*
Extra Attack, however, went a bit too far IMO by allowing Monks to have it, so 5 out of 6 martial classes get it!

As I said in other posts, spells are a huge offender IMO and so I reworked the spell lists to create more unique spells.
Saving throws were just a side note in the process of reworking our 5E Mod. When I saw Clerics, Paladins, and Warlock all had the same saves, I thought "This is crazy? If they are so similar, why bother making them distinct classes?" Frankly, you could combine them all into a core class, Devotee or something, and make each distinct subclasses. But, in an effort to keep them distinct, I reassigned them new saves to decrease the areas they overlapped a bit more.
I wouldn't mind seeing different mechanics for casters instead of identical spell slot progress for Bards, Clerics, Druids, Sorcerers, and Wizards. Like Extra Attack for martials, 5 out of 6 casting classes progress nearly identically. Only Warlocks are currently a stand out, which I believe is one reason why so many find the class appealing.
Within a class is where flavor, fluff, feel for a character is largely about your concept IMO, not the mechanics behind it. That might mean the mark was missed on the design for core classes a bit, with gaps that players are finding, but making more of everything the same for everyone only aggravates the problem and at that point, just get rid of classes because they no longer serve a purpose of defining what is unique about your PC compared to other classes.
*Tasha, of course, ruined this by allowing a Fighting Style as a feat. BOO! Bad form!
