Does 5e's magic feel same-y to you? The spellcasters?
Unfortunately yes, but I don't have a good solution. I prefer each class to bring something unique to the table so the player gets a chance to shine. Mixing together magic so everyone can do basically the same thing is part of why I avoided 4E, and I'm not a fan of seeing the same effect reskinned (save or take 1d8 damage, attack and do 1d10 damage on a hit). It's why I'm not a fan of the Bard. I originally liked the idea of a full-casting class, but then I saw the list (wizard spells minus the boom-boom). That's "meh." Perhaps the solution is simply a smaller, but unique, spell list that interplays with the class features, such as the Paladin got.
However, this can swing the other way. I initially wasn't a fan of anyone, not just the rogue, finding and disabling traps, though now I understand the idea: rogues used to be mandatory in a group, and we don't want to force someone into a mandatory role. So, I get it.
Cantrips: I'm not a fan of the damaging ones. Cantrips originally were nuisance spells with occasional strategic use, and now, they've morphed into unlimited primary damage abilities that eventually surpass slotted spells. While I understand the idea - let casters contribute every round in combat like the fighter does because sitting around doing nothing or using a weapon that sucks so bad you'll never hit stinks - there was a cost: blending of the classes into feeling somewhat the same.
And do you like the design principle (spell equivalent) of 5e?
As to spell-like abilities for various classes, I'm okay. Barbarians probably should be supernatural. If the only way to express this is to mimic an existing spell, so be it.