In his cover of neon genesis evangelon opening, he does something throughout the song at 0:21 and other times throughout the verse. It sounds almost like a thumb slap but doing just a thumb slap doesnt get the same sound.
Throughout the song, he is using a finger-style technique with hybrid picking. The specific technique that he uses beginning at 0:18 (which you can start to see around 0:21) is called a thumb slap, as you mentioned in the question itself and as advised by the commenters.
The thumb slap is the finger-style version of the Slap Strum technique, which is used to add a percussive, 'snare' sound to the guitar part.
If you are attempting to precisely replicate the exact sound that he is making in the video, then ensure that you are using a thumb-pick, firstly. His is plastic so, yours should be as well; and, you may want to even attempt to somehow match the thickness. Secondly, practice your thumb-slapping on a guitar with an acrylic face/body, as the performer is using (or purchase the exact same guitar.) Finally, really mute that bottom-string after you slap it, like he is doing...
I can't imagine why "doing just a thumb slap doesnt get the same sound," as you put it. It looks and sounds like a fairly straight-forward thumb slap, in truth...
At 0:25 (OP edited the question from 0:21 at first,) the performer is utilizing a strumming style called Rasgueado that is commonly used in flamenco style guitar.