For pure delay...there's really no difference using a delay effect/plugin VS duplicating/sliding one track.
It's IMO actually cleaner with the copy/delay method, as there is no need to add plugs/processing, it's just a straight playback of another track.
Now...if you wanted some kind of repeating delay with feedback, etc...then using an effect/plug is probably easier.
AFA as how it sounds....it sounds like a track with a single delay. There's no good or bad about it....it's all up to what you want for the mix. I've agreed that if what you really want is double tracked vocals....then double track the vocals.
That said, there are times when a perfect copy with delay might actually sound just right for the mix...because, you get the exact/perfect L/R delay bounce.
I've used the copy/delay approach on other things two, not just vocals...but same thing with instruments as with vocals, if you prefer to double track, then double track. Sometimes I prefer to double track, sometimes I copy/delay.
It's just production choices...there's no right/wrong about it. Try it both ways, listen to what yo have...pick one or the other.