Further, any kit can sound like that with enough processing.
Don't forget tuning, head choice, muffling, and porting. Replace that kit with a tightly-tuned, unmuffled, un-ported bop kit with single ply heads, and you'd get something entirely different.
The hoops are also contributing somewhat to the dry, focused tom sound, so the OP might want to consider a kit with die-cast hoops. Toms are tuned low, of course, and this is appropriate for some genres. Low tuning, muffling, bass drum ports -- these are not bad or good; it depends on the music you wish to play, and how you want to sound to the audience.
But the processing on this recording is considerable.
At minimum, the processing requirements would be an EQ, Gate, and compressor for every channel/mic, a stereo pair of overheads, and at least one room mic a bit further away from the kit. Most hardware mixers don't have the processing, so you'd need at least a Presonus Studiolive (or similar), or an 8-channel preamp and a computer running Logic, Protools, etc.
But, you really do need the large, reverberant room, to get that sound. Even the most expensive reverb software sounds pretty lame, when compared to real sound waves bouncing around the place, captured by microphones (preferably a stereo pair). The very best software and gadgets around are not very convincing, if you've been lucky enough to record the real thing.
It's not uncommon to search out a good-sounding room to record drums. Zeppelin famously hung microphones off a balcony to record Levy Breaks. Phil Collins used a room with a stone wall. The Red Hot Chili Peppers used the marble foyer of a mansion. And so on. It's about giving the drum sound some character. But in this case, the character is really coming from the room and the processing. Muffling, tuning, head choice, and the bass drum head port are also playing a role.
But the OP's ears are probably hearing that reverb, mostly. Which isn't *really* the drum sound.