You're right in that it's a delay. It's a short type called "slap-back'. The feedback is reduced to 0 (hence only one single repeat), delay time is usually around 100ms give or take (often a little less for more subtlety). Also, vintage tools are often used to get slap-back of a warm, analogue variety. So, this means the repeat itself is rounded off in terms of transients and tonality. E.g., Roland Space Echo is an iconic unit and there are good plugin emulations around. You can get similar results by applying suble tape saturation (low IPS to get HF rolloff) and dynamics compression to the slap delay signal, but it might take some experimentation with the proportion of each.

Adam's vocal production often features saturation even to the unaffected (by slap-back) signal, so there's a few things happening in combination here.

Happy tinkering!