But, as far as magic goes, when no one has a schtick, then, it becomes commonplace and humdrum. Fighter walks up, Action Surges and burns a couple of Superiority Dice and everyone knows that that's a Battlemaster. Character walks up and drops Fireball, he could be a cleric, druid, warlock, sorcerer or wizard. There's nothing special there.
I've been pondering just what it means for something to feel magical, and I think this is a lot of it. My first experience with D&D was in the early/mid 80's, and let me tell you, having Bargle cast a Magic Missile and kill my cleric friend definitely felt like there was magic going on. That was literally more than 30 years ago (when did I get so old?!) and while I don't think the implementation is worse, the "magic" is somewhat gone for me because I know the tropes and have developed a pretty substantial amount of system mastery/familiarity with D&D overall. Fireball doesn't feel like magic to me anymore, it feels like a game mechanic -- but that's not really a complaint nor a criticism of 5e magic.
I think a lot of the feeling of "magic" has to do with uniqueness of abilities, like Hussar said, but I think that that uniqueness can be implemented either mechanically or thematically. For example, in the game I'm currently playing in, we have a gnome wizard who walks around throwing fireballs. He doesn't feel particularly magical to me; he feels more like a walking artillery piece. On the other hand, my character (a Ftr 1/Warlock 5) is conceptually a non-magical warrior who had demons channeled into him unwillingly during a ritual. He can control the demons to an extent, resulting in demonic energy bursting out from him (i.e., casting warlock spells) but mostly is a melee warrior-type. Even though the mechanics of "Ftr 1/Warlock 5" don't feel particularly magical, in play, my character does feel pretty magical.
What I've found in 5e, somewhat paradoxically, is that the primary casters don't feel very magical to me, but gish-types do. 5e paladins channeling spell slots into smites feels much more magical to me than the 3e/PF implementation of smiting, however much the end result is the same. Eldritch knights, valor bards, bladelocks? Yep, all of them. I think it's because (barring 4e, which I didn't really play) being a gish has never been super-well supported in the core rules. In a world where magic is a tool, it makes sense that some number of fighter-types would have figured out specialized magical tricks to use in combat, and 5e feels like it may showcase that better than any previous edition. Meanwhile, wizards and clerics are kind of old hat -- regardless of how much they focus on spellcasting, they don't feel unique and magical anymore.