I always thought that was called "Okinawan" style... I first liked it in Seagal's movies. Was so quick and brutal. (and had to make the attacker just think he was screwed...). Everything countered so quickly and effectively. It's why it works so well on film.
I think this is mostly what he's into these days, but in his early films, it was the hit fast, hit first, hit hard Okinawan style he used. Visually, it's just impressive....for morale, it's simply demoralizing to see a guy go down that fast and hard.... I imagine the reflexes as we get older though, aren't up to it (so switching to more kicks vs. punches, etc.)
Yes It's Aikido Seagal uses. It's more aggressive than what you see here state side. His Japanese instructor Sensei Tohei was known for his hard style Aikido, Sensei Tohei taught it a lot closer to its relative Daito Ryu Aiki Jutsu. Kali is a good martial system but it is too formulaic in my opinion. Sensei (Take) Seagal does not hold any official rank in an Okinawan art form. His main focus is Aikido in which he is ranked to 7th dan and some mixed in Wing Chun.
This is his first movie and opening scene. In it he is displaying classical Aikikai Aikido. The close-line move that he is known for is called Irimi Nage.
Weird, will have to find the interview I'm recalling with him. Of course, I could be remembering it wrong too. It was an interview shortly after he did Above The Law.