That's simply not true.
The only thing that distinguishes a 3e elf from a half elf is a couple of skills. Elves gain +2 on Listen, Search and Spot while Half-elves are given +2 on Diplomacy and Gather Information. Oh, and elves get a couple of weapon proficiencies that most classes get anyway.
Wow, they are so truly distinct and unique. Good grief, it specifically says that half elves are considered an elf.
But it all comes down to how you define "unique". In your 3e game, all half elves are identical. They all have exactly the same traits. You cannot have a half elf that doesn't share the same traits as every other half elf in the PHB.
In my 5e (2024) game though, I could have five half-elves in the group and not a single one shares a single trait. How's that for unique? Not a single one of them would share a single trait. I'd say that that's far more unique.
I can play a traditional half-elf simply by using the "elf" traits. It's virtually identical to the 5e (2014) version. As in, AFAIR, the only change would be a couple of proficiencies. Or, I can go wild, have a half elf half silver dragon, use the Silver Dragonborn traits and poof, I'm good to go.
Unique races for the win.